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Cengage Maths Solutions Straight Lines - Coordinate Geometry

Cengage Maths Solutions Class 12 Straight Lines (https://doubtnut.com/cengage-solutions/chapter-straight-lines-1) PDfs provides accurate solutions to practice before your final exam. Students can study this comprehensive Cengage Maths PDf Solutions on Coordinate Geometry Straight Lines and its various sub-topics for effective exam preparation. The Cengage Class 12 PDF solutions help the student to swiftly practice and revise the important concepts just before the annual exam to score high marks in maths.<br>For more such video solutions/Pdf solution, visit https://doubtnut.com and clear your doubt instantly.

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Cengage Maths Solutions Straight Lines - Coordinate Geometry

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  1. CENGAGE / G TEWANI MATHS SOLUTIONS CHAPTER GEOMETRY STRAIGHT LINES || COORDINATE  Download Doubtnut Today Ques No. Question CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the equation of the line through (2,3) which is (i) parallel to the x-axis and (ii) parallel to the y-axis. 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the equation of the line which intersects the y-axis at a distance of 2 units above the origin and makes an angle of 300 with the positive direction of the x-axis. 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the equation of a straight line cutting off an intercept-1 from the y-axis and being equally inclined to the axes. 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the equation of a line that has -y-intercept 4 and is a perpendicular to the line joining (2, - 3) and (4,2). 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the equation of the perpendicular bisector of the line segment joining the points A(2, 3) and B(6, - 5). 5  Watch Free Video Solution on Doubtnut

  2. CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the equation of the line which is parallel to 3x - 2y + 5 = 0 and passes through the point (5, - 6) 6  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the equation of the straight line that passes through the point (3, 4) and is perpendicular to the line 3x + 2y + 5 = 0 7  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different In what ratio does the line joining the points (2, 3) and (4, 1) divide the segment joining the points (1, 2) and (4,3)? 8  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the locus of a point P which moves such that its distance from the line y =√3x - 7 ( ) 9 is the same as its distance from 2√3, - 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Consider a triangle with vertices A(1, 2), B(3,1), and C( - 3,0). Find the equation of altitude through vertex A. the equation of median through vertex A. the equation of internal angle bisector of ∠A. 10

  3.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different A rectangle has two opposite vertices at the points (1, 2) and (5,5). It these vertices lie on the line x = 3 , find the other vertices of the rectangle. 11  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different A line is such that its segment between the lines 5x - y + 4 = 0 and 3x + 4y - 4 = 0 is bisected at the point (1,5). Obtain its equation. 12  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different ABCD is a square whose vertices are A(0,0),B(2, 0), C(2,2), and D(0,2) . The square is rotated in the XY - plane through an angle 300 in the anticlockwise sense about an axis passing though A perpendicular to the XY - plane . Find the equation of the diagonal BD of this rotated square. 13  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different In a triangle ABC, side AB has equation 2x + 3y = 29 and side AC has equation x + 2y = 16. If the midpoint of BC is 5, 6), then find the equation of BC. 14  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Different

  4. Forms Of Line 15 Two consecutive sides of a parallelogram are 4x + 5y = 0 and 7x + 2y = 0 . If the equation of one diagonal is 11x = 7y = 9, find the equation of the other diagonal.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the coordinates of the foot of the perpendicular drawn from the point (1, - 2) on the line y = 2x + 1. 16  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different 17 Find the image of the point ( - 8,12) with respect to the line mirror 4x + 7y + 13 = 0 .  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different If one of the sides of a square is 3x - 4y - 12 = 0 and the center is (0, 0) , then find the equations of the diagonals of the square. 18  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different A vertex of an equilateral triangle is (2, 3) and the equation of the opposite side is x + y = 2. Find the equation of the other sides of the triangle. 19  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Different

  5. Forms Of Line A line 4x + y = 1 through the point A(2, - 7) meets the line BC whose equation is 3x - 4y + 1 = 0 at the point B. Find the equation of the line AC, so that AB = AC. 20  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different A ray of light is sent along the line x - 2y - 3 = 0 . Upon reaching the line 3x - 2y - 5 = 0 , the ray is reflected. Find the equation of the line containing the reflected ray. 21  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different A ray of light is sent along the line 2x - 3y = 5. After refracting across the line x + y = 1 it enters the opposite side after torning by 150 away from the line x + y = 1 . Find the equation of the line along which the refracted ray travels. 22  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the equation of the line passing through the point (2, 2) and cutting off intercepts on the axes whose sum is 9. 23  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the equation of the straight line that passes through the point ( - 5, 4) and is such that the portion intercepted between the axes is divided by the point in the ratio 1:2. 24  Watch Free Video Solution on Doubtnut

  6. CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Reduce the line 2x - 3y + 5 = 0 in slope-intercept, intercept, and normal forms. 25  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the equation of the line which is at a perpendicular distance of 5 units from the origin and the angle made by the perpendicular with the positive x-axis is 300. 26  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different A straight canal is 41 2 miles from a placce and the shortest route from this place to the canal is exactly north-east. A village is 3 miles north and 4 miles east from the place. Does it lie by the nearest edge of the canal? 27  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different A variable straight line is drawn through the point of intersection of the straight lines x a+ a locus of the midpoint of AB is the curve 2xy(a + b) = ab(x + y) y b= 1 and x y 28 = 1 and meets the coordinate axes at A and B. Show that the b+  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different ( ) ( ) ( ) ( ) ( ) x a y b 1 a2 1 1 c2 , where c is If the line = 1 moves in such a way that + + = b2 29 a constant, prove that the foot of the perpendicular from the origin on the straight line describes the circle x2+ y2= c2.  Watch Free Video Solution on Doubtnut

  7. CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the equation of the right bisector of the line segment joining the points (3, 4) and ( - 1, 2). 30  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different If the coordinates of the points A, B,C and D be (a, b),(a′,b′ ), ( - a,b) and (a′, - b′) respectively, then the equation of the line bisecting the line segments AB and CD is 31  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different If the coordinates of the vertices of triangle ABC are ( - 1, 6),( - 3, - 9) and (5, - 8) , respectively, then find the equation of the median through C. 32  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the equation of the line perpendicular to the line x y b= 1 and passing through a a- 33 point at which it cuts the x-axis.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different If the middle points of the sides BC, CA, and AB of triangle ABC are (1, 3), (5, 7), and 34

  8. ( - 5, 7), respectively, then find the equation of the side AB.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the equations of the lines which pass through the origin and are inclined at an angle tan-1m to the line y = mx + ⋅ 35  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different If ( - 2,6) is the image of the point (4, 2) with respect to line L = 0 , then find the equation of line L. 36  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different If the lines x + (a - 1)y + 1 = 0 and 2x + a2y - 1 = 0 are perpendicular, then find the value of a. 37  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different 38 Find the area bounded by the curves x + 2|y| = 1 and x = 0 .  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different

  9. Find the equation of the straight line passing through the intersection of the lines Find the equation of the straight line passing through the intersection of the lines x - 2y = 1 and x + 3y = 2 and parallel to 3x + 4y = 0. 39  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different A straight line through the point (2, 2) intersects the lines √3x + y = 0 and √3x - y = 0 at the point A and B, respectively. Then find the equation of the line AB so that triangle OAB is equilateral. 40  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different A straight line through the point A(3,4) is such that its intercept between the axis is bisected at A. Find its equation. 41  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different The diagonals AC and BD of a rhombus intersect at (5, 6). If A ≡ (3,2), then find the equation of diagonal BD. 42  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different If the foot of the perpendicular from the origin to a straight line is at (3, - 4) , then find the equation of the line. 43  Watch Free Video Solution on Doubtnut

  10. CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different 44 If we reduce 3x + 3y + 7 = 0 to the form xcosα + ysinα = p, then find the value of p.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the equation of the straight line which passes through the origin and makes angle 600 with the line x +√3y +√3 = 0 . 45  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different A line intersects the straight lines 5x - y - 4 = 0 and 3x - 4y - 4 = 0 at A and B , respectively. If a point P(1, 5) on the line AB is such that AP:PB = 2:1 (internally), find point A. 46  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the equations of the lines, which cut-off intercepts on the axes whose sum and product are 1 and -6 , respectively. 47  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Two fixed points A and B are taken on the coordinates axes such that OA = a and OB = b . Two variable points A′ and B′ are taken on the same axes such that OA′+ OB′= OA + OB. Find the locus of the point of intersection of AB′ and A′B. 48

  11.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different The perpendicular from the origin to a line meets it at the point ( - 2, 9). Find the equation of the line. 49  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the direction in which a straight line must be drawn through the point -1,2) so that its point of intersection with the line x + y = 4 may be at a distance of 3 units from this point. 50  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the equation of the lines through the point (3, 2) which make an angle of 450 with the line x - 2y = 3. 51  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Form Of A Line GEOMETRY_STRAIGHT LINES_Distance A straight line is drawn through the point P(2,3) and is inclined at an angle of 300 with the x-axis. Find the coordinates of two points on it at a distance 4 from P. 52  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Form Of A Line GEOMETRY_STRAIGHT LINES_Distance Find the equation of the line passing through the point A(2,3) and making an angle of 450 with the x-axis. Also, determine the length of the intercept on it between A and the line x + y + 1 = 0 53  Watch Free Video Solution on Doubtnut

  12. CENGAGE_MATHS_COORDINATE Form Of A Line GEOMETRY_STRAIGHT LINES_Distance The line joining two points A(2, 0) and B(3, 1) is rotated about A in anticlockwise direction through an angle of 150 . Find the equation of the line in the new position. If B goes to C in the new position, what will be the coordinates of C? 54  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Form Of A Line GEOMETRY_STRAIGHT LINES_Distance Find the distance of the point (1, 3) from the line 2x - 3y + 9 = 0 measured along a line x - y + 1 = 0 55  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Form Of A Line GEOMETRY_STRAIGHT LINES_Distance 56 Two adjacent vertices of a square are (1, 2) and ( - 2, 6) Find the other vertices.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Form Of A Line GEOMETRY_STRAIGHT LINES_Distance A line through the variable point A(k + 1, 2k) meets the lines 7x + y - 16 = 0,5x - y - 8 = 0,x - 7x + y − 16 = 0,5x − y − 8 57 = 0,x − 5y + 8 = 0 at B,C,D, respectively. Prove that AC,AB, AD are in HP.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Form Of A Line GEOMETRY_STRAIGHT LINES_Distance Two particles start from the point (2, - 1); one moves 2 unites along the line x + y = 1

  13. 58 and the other 5 units along the line x - 2y = 4 . If the particle move towards increasing distance, then find their new positions.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Form Of A Line GEOMETRY_STRAIGHT LINES_Distance Find the distance between A(2, 3) on the line of gradient 3/4 and the point of intersection P of this line with 5x + 7y + 40 = 0. 59  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Form Of A Line GEOMETRY_STRAIGHT LINES_Distance The center of a square is at the origin and its one vertex is A(2,1). Find the coordinates of the other vertices of the square. 60  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line If p is the length of the perpendicular from the origin to the line x a+y b= 1 , then prove 61 that 1 1 1 b2. p2= a2+

  14.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line Find the coordinates of a point on x + y + 3 = 0, whose distance from x + 2y + 2 = 0 is √5. 62  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line 63 Two sides of a square lie on the lines x + y = 1 and x + y + 2 = 0. What is its area?  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line Prove that the product of the lengths of the perpendiculars drawn from the points √a2- b2,0 and -√a2- b2,0 to the line x ( ) ( ) y bsinθ = 1 is b2. 64 acosθ +  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line Find the least and greatest values of the distance of the point (cosθ,sinθ), θ ∈ R, from the line 3x - 4y + 10 = 0. 65  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line

  15. 66 Find the least value of (x - 2)2+ (y - 2)2 under the condition 3x + 4y - 2 = 0.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line Find the equation of the line which is equidistant from parallel lines 9x + 6y - 7 = 0 and 3x + 2y + 6 = 0. 67  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line If one side of the square is 2x - y + 6 = 0, then one of the vertices is (2, 1) . Find the other sides of the square. 68  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line Prove that the area of the parallelogram contained by the lines 4y - 3x - a = 0,3y - 4x + a = 4y − 3x − a = 0,3y − 4x + a 69 = 0,4y − 3x − 3a = 0, ( ) 2 7 a2. and 3y - 4x + 2a = 0 is  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line Find the equations of straight lines passing through ( - 2, 7) and having an intercept of length 3 between the straight lines 4x + 3y = 12 and 4x + 3y = 3. 70  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line ABC is an equilateral triangle with A(0,0) and B(a, 0) , (a>0). L, M and N are the foot of the perpendiculars drawn from a point P to the side AB,BC,andCA , respectively. If P 71 . lies inside the triangle and satisfies the condition PL2= PM P. PN, then find the locus of  Watch Free Video Solution on Doubtnut

  16. CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line Find the points on y-axis whose perpendicular distance from the line 4x - 3y - 12 = 0 is 3. 72  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line If pandp′ are the distances of the origin from the lines xsecα + ycosecα = kandxcosα - ysinα = xsecα + ycosecα 73 = kandxcosα − ysinα = kcos2α, then prove that 4p2+ p′2= k2.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line ( ) ( ) Prove that the lengths of the perpendiculars from the points m2,2m , mm′,m + m′, ( ) 74 and m′2,2m′ to the line x + y + 1 = 0 are in GP.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line Find the ratio in which the line 3x + 4y + 2 = 0 divides the distance between 3x + 4y + 5 = 0 and 3x + 4y - 5 = 0 75  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of

  17. A Point From A Line 76 Find the equations of lines parallel to 3x - 4y - 5 = 0 at a unit distance from it.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line Find the equation of a straight line passing through the point ( - 5, 4) and which cuts off an intercept of √2 units between the lines x + y + 1 = 0 and x + y - 1 = 0. 77  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Position Of Points Relative To A Line 78 Are the points (3, 4) and (2, - 6) on the same or opposite sides of the line 3x - 4y = 8?  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Position Of Points Relative To A Line Find the set of positive values of b for which the origin and the point (1, 1) lie on the same side of the straight line, a2x + aby + 1 = 0, ∀a ∈ R. 79  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Position Of Points Relative To A Line 80 If the point (a, a) is placed in between the lines |x + y| = 4, then find the values of a.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Position Of Points Relative To A Line

  18. Find the range of values of the ordinate of a point moving on the line x = 1 , and always remaining in the interior of the triangle formed by the lines y = x, the x-axis, and x + y = 4. 81  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Position Of Points Relative To A Line If the point P a2, a lies in the region corresponding to the acute angle between the ( ) 82 lines 2y = x and 4y = x , then find the values of a.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Position Of Points Relative To A Line a2,a + 1 lies in the angle between the lines 3x - y + 1 = 0 and ( ) If the point 83 x + 2y - 5 = 0 containing the origin, then find the value of a.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Position Of Points Relative To A Line The point (8, - 9) with respect to the lines 2x + 3y - 4 = 0 and 6x + 9y + 8 = 0 lies on (a) the same side of the lines (b) the different sides of the line (c) one of the line (d) none of these 84  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Position Of Points Relative To A Line Which pair of points lies on the same side of 3x - 8y - 7 = 0? a) (0, - 1)and(0, 0) b) (4, - 3) and (0, 1) c) ( - 3, - 4)and(1,2) d) ( - 1, - 1)and(3, 7) 85

  19.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Position Of Points Relative To A Line ( 3α 2,a2 lie on the opposite sides of the line ) Find the range of (α, 2 + α) and 86 2x + 3y = 6.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Position Of Points Relative To A Line If the point P a, a2 lies completely inside the triangle formed by the lines x = 0,y = 0, ( ) 87 and x + y = 2, then find the exhaustive range of values of a.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Concurrency Of Three Lines Find the value of λ , if the lines 3x - 4y - 13 = 0,8x - 11y - 33, 3x − 4y − 13 = 0, 8x − 11y 88 − 33, and 2x - 3y + λ = 0 are concurrent.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Concurrency Of Three Lines If the lines a1x + b1y + 1 = 0,a2x + b2y + 1 = 0, a1x + b1y + 1 = 0,a2x + b2y + 1 = 0, 89 ( ) ( ) and a3x + b3y + 1 = 0 are concurrent, then show that the points a1,b1, a2,b2 are collinear.  Watch Free Video Solution on Doubtnut

  20. CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations OfCENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines For the straight lines 4x + 3y - 6 = 0 and 5x + 12y + 9 = 0, find the equation of the bisector of the obtuse angle between them, bisector of the acute angle between them, and bisector of the angle which contains (1, 2) 90  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines Two equal sides of an isosceles triangle are given by 7x - y + 3 = 0 and x + y = 3, and its third side passes through the point (1, - 10). Find the equation of the third side. 91  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines The vertices BandC of a triangle ABC lie on the lines 3y = 4xandy = 0 , respectively, ( . If ABOC is a rhombus lying in the ) 2 3,2 92 and the side BC passes through the point 3 first quadrant, O being the origin, find the equation of the line BC.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines ( If the line ax + by = 1 passes through the point of intersection of y = xtanα + psecα, ysin 30∘ y = xtanα + psecα, ysin(30∘− α) − xcos(30∘− α) 93 = p 3 , and is inclined at 30∘ with y = tanαx , then prove that a2+ b2= 4p2 .  Watch Free Video Solution on Doubtnut

  21. CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations OfCENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines 94 Find the image of the point (4, - 13) in the line 5x + y + 6 = 0.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines 95 Find the foot of the perpendicular from the point (2,4) upon x + y = 1.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines In △ ABC, vertex A is (1, 2). If the internal angle bisector of ∠B is 2x - y + 10 = 0 and the perpendicular bisector of AC is y = x, then find the equation of BC 96  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines Find the equation of the bisector of the obtuse angle between the lines 3x - 4y + 7 = 0 and 12x + 5y - 2 = 0. 97  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines The incident ray is along the line 3x - 4y - 3 = 0 and the reflected ray is along the line 24x + 7y + 5 = 0. Find the equation of mirrors. 98  Watch Free Video Solution on Doubtnut

  22. CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations OfCENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines If the two sides of rhombus are x + 2y + 2 = 0 and 2x + y - 3 = 0, then find the slope of the longer diagonal. 99  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines In triangle ABC , the equation of the right bisectors of the sides AB and AC are x + y = 0 and y - x = 0 , respectively. If A ≡ (5, 7) , then find the equation of side BC. 100  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Family Of Straight Lines Show that the straight lines x(a + 2b) + y(a + 3b = a + b for different values of a and b pass through a fixed point. Find the point. 101  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Family Of Straight Lines Let ax + by + c = 0 be a variable straight line, whre a,bandc are the 1st, 3rd, and 7th terms of an increasing AP, respectively. Then prove that the variable straight line always passes through a fixed point. Find that point. 102  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Family Of Straight Lines Prove that all the lines having the sum of the interceps on the axes equal to half of the product of the intercepts pass through the point. Find the fixed point. 103  Watch Free Video Solution on Doubtnut

  23. CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Family Of Straight Lines Find the straight line passing through the point of intersection of 2x + 3y + 5 = 0, 5x - 2y - 16 2x + 3y + 5 = 0,5x − 2y − 16 104 = 0 , and through the point ( - 1,3).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Family Of Straight Lines Consider a family of straight lines (x + y) + λ(2x - y + 1) = 0 . Find the equation of the straight line belonging to this family that is farthest from (1, - 3). 105  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Family Of Straight Lines If 5a + 5b + 20c = t, then find the value of t for which the line ax + by + c - 1 = 0 always passes through a fixed point. 106  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Family Of Straight Lines Find the values of non-negative real number h1,h2, h3, k1, k2, k3 such that the algebraic ( ) ( ) ) ( ) ( ) ( sum of the perpendiculars drawn from the points 2,k1, 3, k2, ⋅ 7, k3, h1, 4 , h2,5 , 107 (2,k1),(3, k2), ⋅ 7,k3),(h1,4), (h2,5), (h3, − 3) on a variable line passing through (2, 1) is zero.  Watch Free Video Solution on Doubtnut

  24. CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Family Of Straight Lines If aandb are two arbitrary constants, then prove that the straight line (a - 2b)x + (a + 3b)y + 3 (a − 2b)x + (a + 3b)y + 3a 108 + 4b = 0 will pass through a fixed point. Find that point.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Family Of Straight Lines (( )) ( ) ( ) x a y b l c If a, b,c are in harmonic progression, then the straight line + + = 0 109 always passes through a fixed point. Find that point.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Family Of Straight Lines If the algebraic sum of the distances of a variable line from the points (2,0),(0, 2), and ( - 2, - 2) is zero, then the line passes through the fixed point. (a)( - 1, - 1) (b) (0, 0) (c) (1, 1) (d) (2, 2) 110  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Family Of Straight Lines Consider the family of lines 5x + 3y - 2 + λ1(3x - y - 4) = 0 5x + 3y − 2 + λ1(3x − y − 4) 111 = 0 and x - y + 1 + λ2(2x - y - 2) = 0 . Find the equation of a straight line that belongs to both the families.  Watch Free Video Solution on Doubtnut

  25. CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Family Of Straight Lines If x y m= 1 is any line passing through the intersection point of the lines x y a= 1 then prove that 1 a+y l+ b= 1 112 and x 1 m= 1 a+ 1 b b+ l+  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines Show that the reflection of the line ax + by + c = 0 on the line x + y + 1 = 0 is the line b + ay + (a + b - c) = 0 where a ≠ b. 113  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Family Of Straight Lines Straight lines y = mx + c1 and y = mx + c2 where m ∈ R+, meet the x-axis at A1andA2, respectively, and the y-axis at B1andB2, respectively. It is given that points A1,A2, B1, and B2 are concylic. Find the locus of the intersection of lines A1B2 and A2B1 . 114  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Position Of Points Relative To A Line A line L1≡ 3y - 2x - 6 = 0 is rotated about its point of intersection with the y-axis in the clockwise direction to make it L2 such that the are formed by L1, L2 the x-axis, and line x = 5 is 49 3sq equation of L2. . 115 units if its point of intersection with x = 5 lies below the x-axis. Find the

  26.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Family Of Straight Lines Find the locus of the circumcenter of a triangle whose two sides are along the coordinate axes and the third side passes through the point of intersection of the line ax + by + c = 0 and lx + my + n = 0. 116  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different A regular polygon has two of its consecutive diagonals as the lines √3x + y -√3 and 2y =√3 . Point (1,c) is one of its vertices. Find the equation of the sides of the polygon and also find the coordinates of the vertices. 117  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Form Of A Line GEOMETRY_STRAIGHT LINES_Distance A diagonal of rhombus ABCD is member of both the families of lines (x + y - 1) + λ(2x + 3y - (x + y − 1) + λ(2x + 3y − 2) = 0 and (x - y + 2) + λ(2x - 3y + 5) = 0 (x − y + 2) + λ(2x − 3y + 5) 118 = 0 and rhombus is (3, 2). If the area of the rhombus is 12√5 sq. units, then find the remaining vertices of the rhombus.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different A triangle has two sides y = m1x and y = m2x where m1 and m2 are the roots of the equation bα2+ 2hα + a = 0. If (a, b) be the orthocenter of the triangle, then find the equation of the third side in terms of a, b and h. 119  Watch Free Video Solution on Doubtnut

  27. CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Position Of Points Relative To A Line Let A ≡ (6, 7), B ≡ (2,3)andC ≡ ( - 2, 1) A ≡ (6, 7),B ≡ (2,3)andC 120 ≡ ( − 2,1) be the vertices of a triangle. Find the point P in the interior of the triangle such that PBC is an equilateral triangle.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Position Of Points Relative To A Line ( ) Find all the values of θ for which the point sin2θ,sinθ lies inside the square formed 121 by the line xy = 0 and 4xy - 2x - 2y + 1 = 0.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Position Of Points Relative To A Line The equations of two sides of a triangle are 3y - x - 2 = 0andy + x - 2 = 0. 3y − x − 2 = 0andy + x − 2 122 = 0. The third side, which is variable, always passes through the point (5, - 1) . Find the range of the values of the slope of the third side, so that the origin is an interior point of the triangle.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Let ABC be a given isosceles triangle with AB = AC . Sides ABandAC are extended up to EandF, respectively, such that BExCF = AB2. Prove that the line EF always passes through a fixed point. 123  Watch Free Video Solution on Doubtnut

  28. CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Let L1= 0andL2= 0 be two fixed lines. A variable line is drawn through the origin to . P. is a point on the line AB such that (m + n) m OR+ n . cut the two lines at R and S = OP OS 124 Show that the locus of P is a straight line passing through the point of intersection of ) the given lines R, S,R are on the same side of O .  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Concurrency Of Three Lines n 1 ∑ i=1 A variable line cuts n given concurrent straight lines at A1,A2...An such that is OAi 125 a constant. Show that A,A , A such it always passes through a fixed point, O being the point of intersection of the lines  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines Two sides of a rhombus lying in the first quadrant are given by 3x - 4y = 0 and 12x - 5y = 0. If the length of the longer diagonal is 12, then find the equations of the other two sides of the rhombus. 126  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different If D,E, andF are three points on the sides BC,AC,andAB of a triangle ABC such that AD, BE, andCF are concurrent, then show that BD ⋅ CE ⋅ AF = DC ⋅ EA ⋅ FB . 127

  29.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line Consider two lines L1andL2 given by x - y = 0 and x + y = 0 , respectively, and a ( ) moving point P(x, y). Let d P, L1, i = 1,2, represents the distance of point P from the line Li. If point P moves in a certain region R in such a way that ( ) ( ) 128 ≤ 4 , find the area of region R. 2 ≤ d P, P1+ d P, L1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines Consider two lines L1andL2 given by a1x + b1y + c1= 0anda2x + b2y + c2= 0respectivelywher a1x + b1y + c1 = 0anda2x + b2y + c2 = 0respectivelywherec1 and c2 ≠ 0, 129 . intersecting at point P AandB, respectively, such that PA = PB . Similarly, one more line L4 is drawn through the origin meeting the lines L1andL2 at A1andB2, respectively, such that PA1= PB1. Obtain the combined equation of lines L3andL4. A line L3 is drawn through the origin meeting the lines L1andL2 at  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Show that if any line through the variable point A(k + 1, 2k) meets the lines 7x + y - 16 = 0,5 7x + y − 16 = 0,5x − y − 8 130 = 0,x − 5y + 8 = 0 at B,C,D, respectively, the AC, AB, andAD are in harmonic progression. (The three . ). lines lie on the same side of point A  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Show that the lines 4x + y - 9 = 0,x - 2y + 3 = 0,5x - y - 6 = 0 4x + y − 9 = 0, x − 2y + 3 131 = 0,5x − y − 6 = 0 make equal intercepts on any line of slope 2.

  30.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Having given the bases and the sum of the areas of a number of triangles which have a common vertex, show that the locus of the vertex is a straight line. 132  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the locus of the point at which two given portions of the straight line subtend equal angle. 133  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Find the equations of the diagonals of the square formed by the lines x = o,y = 0, x = 1andy = 1. 134  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different ( ) ( - 2,y1 lies on the line joining the points ) ( ) If each of the points x1, 4 , 135 (2, - 1)and(5, - 3) , then the point P x1, y1 lies on the line. (a) 6(x + y) - 25 = 0 (b) 2x + 6y + 1 = 0 (c)2x + 3y - 6 = 0 (d) 6(x + y) + 25 = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different

  31. c2 (ab) (b) 2c2 The area of a parallelogram formed by the lines ax ± by ± c = 0 is (a) (ab) (c) 136 c2 2ab (d) none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE LINES_Miscellaneous GEOMETRY_STRAIGHT The straight line ax + by + c = 0 , where abc ≠ 0, will pass through the first quadrant if (a) ac > 0,bc > 0 (b) ac > 0 or bc < 0 (c)bc > 0 or ac > 0 (d) ac < 0 or bc < 0 137  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different A square of side a lies above the x-axis and has one vertex at the origin. The side ( ) π 4 passing through the origin makes an angle α 0 < α < with the positive direction of x-axis. equation its diagonal not passing through origin is (a) y(cosα + sinα) + x(sinα - cosα) = y(cosα + sinα) + x(sinα − cosα) = α (b)y(cosα + sinα) + x(sinα + cosα) = α y(cosα + sinα) 138 + x(sinα + cosα) = α

  32. (c)y(cosα + sinα) + x(cosα - sinα) = α y(cosα + sinα) + x(cosα − sinα) = α (d)y(cosα - sinα) - x(sinα - cosα) = α y(cosα − sinα) − x(sinα − cosα) = α  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line If the sum of the distances of a point from two perpendicular lines in a plane is 1, then its locus is a square (b) a circle a straight line (d) two intersecting lines 139  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE LINES_Miscellaneous GEOMETRY_STRAIGHT ABC is a variable triangle such that A is (1, 2), and BandC on the line y = x + λ(λ is a variable). Then the locus of the orthocentre of DeltaABC is x + y = 0 (b) x - y = 0 x2+ y2= 4 (d) x + y = 3 140  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE LINES_Miscellaneous GEOMETRY_STRAIGHT The area enclosed by 2|x| + 3|y| ≤ 6 is (a)3 sq. units (b) 4 sq. units (c)12 sq. units (d) 24 sq. units 141  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different The lines y = m1x, y = m2xandy = m3x make equal intercepts on the line x + y = 1. Then ( )( ) ( )( ) (a) 2 1 + m1 1 + m3= 1 + m2 2 + m1+ m3 2(1 + m1)(1 + m3) = (1 + m2)(2 + m1+ m3) ( )( = 1 + m2 ) ( )( ) (b) 1 + m1 1 + m3 1 + m1+ m3 (1 + m1)(1 + m3) = (1 + m2)(1 + m1+ m3) ( 142 1 + m2 )( ) ( 2 + m1+ m3 )( ) (c) 1 + m1 = 1 + m3 (1 + m1)(1 + m2) = (1 + m3)(2 + m1+ m3) ( 1 + m3 )( = 1 + m2 ) ( )( ) (d)2 1 + m1 1 + m1+ m3

  33. 2(1 + m1)(1 + m3) = (1 + m2)(1 + m1+ m3)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different The condition on aandb , such that the portion of the line ax + by - 1 = 0 intercepted between the lines ax + y = 0 and x + by = 0 subtends a right angle at the origin, is a = b (b) a + b = 0a = 2b (d) 2a = b 143  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different One diagonal of a square is along the line 8x - 15y = 0 and one of its vertex is (1, 2). Then the equations of the sides of the square passing through this vertex are 23x + 7y = 9, 7x + 23y = 5323x - 7y + 9 = 0, 7x + 23y + 53 = 0 23x − 7y + 9 = 0, 7x + 23y + 53 = 0 144 23x - 7y - 9 = 0,7x + 23y - 53 = 0 23x − 7y − 9 = 0, 7x + 23y − 53 = 0 none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different If u = a1x + b1y + c1= 0,v = a2x + b2y + c2= 0, u = a1x + b1y + c1= 0, v = a2x + b2y + c2= 0, a1 a2 b2 145 b1 c1 c2 and , then the curve u + kv = 0 is (a)the same straight line u (b)different = =

  34. straight line (c)not a straight line (d)none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE LINES_Miscellaneous GEOMETRY_STRAIGHT The point A(2, 1) is translated parallel to the line x - y = 3 by a distance of 4 units. If the new position A′ is in the third quadrant, then the coordinates of A′ are (a) ( ) ( ) 146 ( ) 2 + 2√2,1 + 2√2 (b) - 2 +√2, - 1 - 2√2 (c) 2 - 2√2, 1 - 2√2 (d) none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different The area of the triangle formed by the lines y = ax, x + y - a = 0 , and the y-axis to a2 2|1 + a| | | 147 1 |1 + a|1 1 a 2|1 + a| (b) (d) 2 1 + a  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different The line x a+y b= 1 meets the x-axis at A, the y-axis at B, and the line y = x at C such, that the area of DeltaAOC is twice the area of DeltaBOC . Then the coordinates of C ( 148 (b) ) ( )( (d) none of these ) b 3,b 2a 3,2a 2b 3,2b are 3 3 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different The equation of a line through the point (1, 2) whose distance from the point (3, 1) has the greatest value is (a)y = 2x (b) y = x + 1 (c)x + 2y = 5 (d) y = 3x - 1 149  Watch Free Video Solution on Doubtnut

  35. CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different A rectangle ABCD has its side AB parallel to line y = x, and vertices A, BandD lie on y = 1,x = 2, and x = - 2, respectively. The locus of vertex C is x = 5 (b) x - y = 5y = 5 (d) x + y = 5 150  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different The centroid of an equilateral triangle is (0, 0). If two vertices of the triangle lie on ( ) x + y = 2√2, then one of them will have its coordinates. (a) √2 +√6,√2 -√6 (b) ( ) ( ) 151 √2 +√3,√2 -√3 (c)√2 +√5,√2 -√5 (d) none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different The lien x 3+y 4= 1 meets the y - and x - axys at AandB , respectively. A square ABCD is constructed on the line segment AB away from the origin. The coordinates of the vertex of the square farthest from the origin are (7, 3) (b) (4, 7) (c) (6, 4) (d) (3, 8) 152  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Position Of Points Relative To A Line A line of fixed length 2 units moves so that its ends are on the positive x-axis and that part of the line x + y = 0 which lies in the second quadrant. Then the locus of the midpoint of the line has equation. (a) x2+ 5y2+ 4xy - 1 = 0 (b) x2+ 5y2+ 4xy + 1 = 0 (c) x2+ 5y2- 4xy - 1 = 0 (d) 4x2+ 5y2+ 4xy + 1 = 0 153  Watch Free Video Solution on Doubtnut

  36. CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Consider the points A(0, 1)andB(2, 0), andP be a point on the line 4x + 3y + 9 = 0 . The ( ) ( ) 12 5,17 -84 5,13 coordinates of P such that |PA - PB| is maximum are (b) - 154 5 5 ( ) 31 7,31 (d) (,0) 7  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different A line L is a drawn from P(4,3) to meet the lines L - 1andL2 given by 3x + 4y + 5 = 0 and 3x + 4y + 15 = 0 at points AandB , respectively. From A , a line perpendicular to L is drawn meeting the line L2 at A1. Similarly, from point B1. Thus, a parallelogram ∀1BB1 is formed. Then the equation of L so that the area of the parallelogram ∀1BB1 is the least is x - 7y + 17 = 07x + y + 31 = 0x - 7y - 17 = 0x + 7y - 31 = 0 155  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Line L has intercepts aandb on the coordinate axes. When the axes are rotated through a given angle keeping the origin fixed, the same line L has intercepts pandq. (b)1 a2+ 1 a2+ q2 1 b2= 1 1 a2+ b2= p2+ q2 (c)a2+ p2= b2+ q2 Then (a) q2 (d) p2+ 156 1 1 b2+ 1 p2=  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Family Of Straight Lines If the straight lines 2x + 3y - 1 = 0, x + 2y - 1 = 0, andax + by - 1 = 0

  37. 2x + 3y − 1 = 0,x + 2y − 1 157 = 0,andax + by − 1 = 0 form a triangle with the origin as orthocentre, then (a,b) is given by (a) (6, 4) (b) ( - 3,3) (c) ( - 8, 8) (d) (0, 7)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE LINES_Miscellaneous GEOMETRY_STRAIGHT Let O be the origin. If A(1,0)andB(0,1)andP(x, y) are points such that xy > 0andx + y < 1, then (a)P lies either inside the triangle OAB or in the third quadrant. (b)P cannot lie inside the triangle OAB (c)P lies inside the triangle OAB (d)P lies in the first quadrant only 158  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different The graph of y2+ 2xy + 40|x| = 400 divides the plane into regions. Then the area of the . units (b) 400sq units (c)800sq 159 . . . bounded region is (a)200sq units (d) 500sq units  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line In a triangle ABC, A = (α,β)B = (2, 3), andC = (1,3). ABC,A = (α,β)B = (2,3), 160 andC = (1,3). Point A lies on line y = 2x + 3, where α ∈ I. The area of ABC, , is such that [Δ] = 5 . The possible coordinates of A are (where [.] represents greatest integer function). (a) (2, 3) (b) (5, 13)( - 5, - 7) (d) ( - 3, - 5)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different a bc- 2 = b c+ c b, where a, b,c > 0, then the family of lines √ax +√by +√c = 0 √ √ If 161 passes though the fixed point given by (1,1) (b) (1, - 2)( - 1,2) (d) ( - 1, 1)  Watch Free Video Solution on Doubtnut

  38. CENGAGE_MATHS_COORDINATE LINES_Miscellaneous GEOMETRY_STRAIGHT P(m,n) (where m,n are natural numbers) is any point in the interior of the quadrilateral formed by the pair of lines xy = 0 and the lines 2x + y - 2 = 0 and 4x + 5y = 20 . The possible number of positions of the point P is. 7 (b) 5 (c) 4 (d) 6 162  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different If the area of the rhombus enclosed by the lines lx ± my ± n = 0 is 2 sq. units, then, a) l,m,n are in G.P b) l,n,m are in G.P. c) lm=n d) ln=m 163  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines In a triangle ABC , the bisectors of angles BandC lies along the lines x = yandy = 0. If A is (1, 2) , then the equation of line BC is (a) 2x + y = 1 (b) 3x - y = 5 (c) x - 2y = 3 (d) x + 3y = 1 164  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different x 2+ y 3= 2 intercepted One of the diagonals of a square is the portion of the line 165 between the axes. Then the extremities of the other diagonal are: (5,5),( - 1,1) (b) (0, 0), (4, 6)(0,0),( - 1, 1) (d) (5, 5), 4,6)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE LINES_Miscellaneous GEOMETRY_STRAIGHT

  39. Two sides of a triangle are along the coordinate axes and the medians through theTwo sides of a triangle are along the coordinate axes and the medians through the vertices (other than the origin) are mutually perpendicular. The number of such triangles is/are zero (b) two (c) four (d) infinite 166  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines In ABC , the coordinates of the vertex A are (4, - 1) , and lines x - y - 1 = 0 and 2x - y = 3 are the internal bisectors of angles BandC . Then, the radius of the encircle 4 √5 (b) √5 167 3 6 7 of triangle ABC is (a) √5 (c) √5 (d)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line Distance of origin from the line 1 +√3 y + 1 -√3 x = 10 along the line y =√3x + k ( ) ( ) 168  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Family Of Straight Lines Distance possible to draw a line which belongs to all the given family of lines y - 2x + 1 + λ1(2y - x - 1) = 0,3y - x - 6 + λ2(y - 3x + 6) = 0,ax + y - 2 + λ3(6x + ay - a) = 0 , then (a)a = 4 (b) a = 3 (c)a = - 2 (d) a = 2 169  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE LINES_Miscellaneous GEOMETRY_STRAIGHT If the equation of any two diagonals of a regular pentagon belongs to the family of lines (1 + 2λ)λ - (2 + λ)x + 1 - λ = 0 and their lengths are sin 360 , then the locus of the

  40. center of circle circumscribing the given pentagon (the triangles formed by these diagonals with the sides of pentagon have no side common) is (a) x2+ y2- 2x - 2y + 1 + sin2720= 0 (b)x2+ y2- 2x - 2y + cos2720= 0 x2+ y2- 2x - 2y + 1 + cos2720= 0 (d)x2+ y2- 2x - 2y + sin2720= 0 170 (c)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE LINES_Miscellaneous GEOMETRY_STRAIGHT ABC is a variable triangle such that A is (1,2) and B and C lie on line y = x + λ (where λ is a variable). Then the locus of the orthocentre of triangle ABC is (a) (x - 1)2+ y2= 4 (b) x + y = 3 (c) 2x - y = 0 (d) none of these 171  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines The locus of the image of the point (2, 3) in the line (x - 2y + 3) + λ(2x - 3y + 4) = 0 is (λ ∈ R) (a) x2+ y2- 3x - 4y - 4 = 0 x2+ y2- 2x - 4y + 4 = 0 (d) none of these 2x2+ 3y2+ 2x + 4y - 7 = 0 172 (b) (c)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different If the intercepts made by the line y=mx by lines y=2 and y=6 is less than 5, then the ( 3⋃ ) ( 4 3,∞ b. ) ( c. ) ( d. none of ) - ∞, -4 -4 3,4 -3 4,4 173 range of values of m is a. 3 3 these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE LINES_Miscellaneous GEOMETRY_STRAIGHT

  41. If P 1 + ( ) t t , 2 + is any point on a line, then the range of the values of t for which √2 √2 174 the point P lies between the parallel lines x + 2y = 1and2x + 4y = 15. is (a) 4√2 3 < t < 5√2 6 (b) 0 < t < 5√2 (c) 4√2 < t < 0 (d) none of these ( ) ( )  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE LINES_Miscellaneous GEOMETRY_STRAIGHT P is a point on the line y + 2x = 1, and QandR two points on the line 3y + 6x = 6 such 2 that triangle PQR is an equilateral triangle. The length of the side of the triangle is √5 175 3 4 (b) √5 (c) √5 (d) none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different The coordinates of two consecutive vertices AandB of a regular hexagon ABCDEF are (1, 0)and(2,0), respectively. The equation of the diagonal CE is (a)√3x + y = 4 (b) x +√3y + 4 = 0 (c)x +√3y = 4 (d) none of these 176  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different The foot of the perpendicular on the line 3x + y = λ drawn from the origin is C. If the line cuts the x and the y-axis at AandB , respectively, then BC:CA is 1:3 (b) 3:1 (c) 1: 9 (d) 9:1 177  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE LINES_Miscellaneous GEOMETRY_STRAIGHT The real value of a for which the value of m satisfying the equation a2- 1 m2- (2a - 3)m + a = 0 given the slope of a line parallel to the y-axis is 3 ( ) 178 2 (b) 0 (c) 1 (d) ±1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different If the quadrilateral formed by the lines

  42. ax + by + c = 0, a′x + b′y + c = 0,ax + by + c′= 0,a′x + b′y + c′= 0 has perpendicular diagonals, then (a) b2+ c2= b′2+ c′2 (b)c2+ a2= c′2+ a′2 (c)a2+ b2= a′2+ b′2 (d) none of these 179  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different The straight lines 7x - 2y + 10 = 0 and 7x + 2y - 10 = 0 form an isosceles triangle with the line y = 2. The area of this triangle is equal to 15 . . . units (b) 10 units18 180 units (d) 7sq 7sq 7sq none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines θ1 and θ2 are the inclination of lines L1andL2 with the x-axis. If L1andL2 pass through ( ) P x1, y1 , then the equation of one of the angle bisector of these lines is (a) x - x1 = (b) = ( y - y1 ( x - x1 ( y - y1 ( x - x1 ( y - y1 ( (c) = ) ) ) ) ) ) θ1- θ2 2 θ1- θ2 2 θ1- θ2 2 θ1-θ2 2 θ1-θ2 2 θ1-θ2 2 cos sin -sin cos sin cos 181 x - x1 ( y - y1 ( (d) = ) ) θ1-θ2 2 θ1-θ2 2 -sin cos  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines A light ray coming along the line 3x + 4y = 5 gets reflected from the line ax + by = 1 and

  43. 182 64 115, b = 112 15 (b)a = 14 15, b = - 8 goes along the line 5x - 12y = 10. Then, (a) a = 115 (c) 64 115,b = - 8 64 15, b = 14 15 115 (d)a = a =  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines Line ax + by + p = 0 makes angle π 4 with xcosα + ysinα = p,p ∈ R+ . If these lines and 183 ( ) the line xsinα - ycosα = 0 are concurrent, then a2+ b2= 1 (b) a2+ b2= 22 a2+ b2 = 1 (d) none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different A line is drawn perpendicular to line y = 5x, meeting the coordinate axes at AandB . If the area of triangle OAB is 10 sq. units, where O is the origin, then the equation of drawn line is 3x - y - 9 (b) x - 5y = 10x + 4y = 10 (d) x - 4y = 10 184  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines If x - 2y + 4 = 0and2x + y - 5 = 0 are the sides of an isosceles triangle having area . units , the equation of the third side is 3x - y = - 9 (b) 3x - y + 11 = 0x - 3y = 19 (d) 3x - y + 15 = 0 185 10sq  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Concurrency Of Three Lines

  44. 186 The number of values of a for which the lines 2x + y - 1 = 0 , ax + 3y - 3 = 0, and 3x + 2y - 2 = 0 are concurrent is 0 (b) 1 (c) 2 (d) infinite  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different The extremities of the base of an isosceles triangle are (2, 0)and(0, 2) . If the equation of one of the equal side is x = 2, then the equation of other equal side is x + y = 2 (b) x - y + 2 = 0y = 2 (d) 2x + y = 2 187  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE LINES_Miscellaneous GEOMETRY_STRAIGHT ( ) ( ) If the equation of the locus of a point equidistant from the points a1, b1 and a2, b2 ( ) ( ) 188 is a1- a2x + b1- b2y + c = 0 , then the value of c is  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Position Of Points Relative To A Line The line PQ whose equation is x - y = 2 cuts the x-axis at P,andQ is (4,2). The line PQ is rotated about P through 450 in the anticlockwise direction. The equation of the line PQ in the new position is (a (a) y = -√2 (b) y = 2 (c) x = 2 (d) x = - 2 189  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different The equation of straight line passing through ( - a, 0) and making a triangle with the axes of area T is (a) 2Tx + a2y + 2aT = 0 (b)2Tx - a2y + 2aT = 0 (c)2Tx - a2y - 2aT = 0 (d)none of these 190  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different If the equation of base of an equilateral triangle is 2x - y = 1 and the vertex is ( - 1,2), 2 20 3 (b) 8 15 2 √ √ √ 191 then the length of the sides of the triangle is √15 15 (d)  Watch Free Video Solution on Doubtnut

  45. CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Family Of Straight Lines The number of integral values of m for which the x-coordinate of the point of intersection of the lines 3x + 4y = 9 and y = mx + 1 is also an integer is 2 (b) 0 (c) 4 (d) 1 192  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different The equation of a straight line on which the length of perpendicular from the origin is four units and the line makes an angle of 1200 with the x-axis is x√3 + y + 8 = 0 x√3 - y = 8x√3 - y = 8 (d) x -√3y + 8 = 0 193  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different ( ) The equation to the straight line passing through the point acos3θ, asin3θ and 194 perpendicular xcosθ + ysinθ = acos2θxsinθ + ycosθ = acos2θ none of these to the line xsecθ + ycosecθ = a is xcosθ - ysinθ = acos2θ  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Family Of Straight Lines

  46. The line parallel to the x-axis and passing through the intersection of the lines ax + 2by + 3b = 0 and bx - 2y - 3a = 0 , where (a,b) ≠ (0, 0) , is (a)above the x-axis at a distance of 3/2 units from it (b)above the x-axis at a distance of 2/3 units from it (c)below the x-axis at a distance of 3/2 units from it (d)below the x-axis at a distance of 2/3 units from it 195  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Concurrency Of Three Lines The straight lines 4ax + 3by + c = 0 , where a + b + c are concurrent at the point a) (4,3) b)(1/4,1/3) c)(1/2,1/3) d) none of these 196  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Concurrency Of Three Lines If lines x + 2y - 1 = 0,ax + y + 3 = 0, and bx - y + 2 = 0 are concurrent, and S is the curve denoting the locus of (a,b) , then the least distance of S from the origin is (a) 5 √57 (b) 5/√51 (c)5/√58 (d) 5/√59 197  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Concurrency Of Three Lines The straight lines x + 2y - 9 = 0, 3x + 5y - 5 = 0 , and ax + by - 1 = 0 are concurrent, if the straight line 35x - 22y + 1 = 0 passes through the point (a, b) (b) (b, a)( - a, - b) (d) none of these 198  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Distance Of A Point From A Line

  47. The coordinates of the foot of the perpendicular from the point (2,3) on the line ( 199 (b) - ) ( (c) ) ( ) ( ) 37 10, - 1 10,37 1 10 37, - 10 (d) 2 3, - 1 3 -y + 3x + 4 = 0 are given by (a) 10 10  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Concurrency Of Three Lines If the extremities of the base of an isosceles triangle are the points (2a,0) and (0, a), and the equation of one of the side is x = 2a, then the area of the triangle is (a) . units (b) 5a2 2 2 units25a2 200 . . 5a2sq units (d) none of these sq sq  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Let P be (5, 3) and a point R on y = x and Q on the x-axis be such that PQ + QR + RP is ( 17 4, 0 (b) (17, 0) (c) ) ( 17 2,0 (d) none of ) 201 minimum. Then the coordinates of Q are (a) these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE LINES_Miscellaneous GEOMETRY_STRAIGHT If sin(α + β)sin(α - β) = sinγ(2sinβ + sinγ), where 0 < α,β, γ < π, then the straight line whose equation is xsinα + ysinβ - sinγ = 0 passes through point (a) (1, 1) (b) ( - 1, 1) (c) (1, - 1) (d) none of these 202  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different . MandN are the variable points of the y-axis such that M lies below A ≡ ( - 4,0),B ≡ (4,0) NandMN = 4 . Lines AMandBN intersect at P. The locus of P is (a) 2xy - 16 - x2= 0 (b) 2xy + 16 - x2= 0 (c) 2xy + 16 + x2= 0 (d)2xy - 16 + x2= 0 203  Watch Free Video Solution on Doubtnut

  48. CENGAGE_MATHS_COORDINATE LINES_Miscellaneous GEOMETRY_STRAIGHT L1andL2 are two lines. If the reflection of L1onL2 and the reflection of L2 on L1 coincide, then the angle between the lines is 300 (b) 600450 (d) 900 204  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines If the straight lines x + y - 2 - 0,2x - y + 1 = 0 and ax + by - c = 0 are concurrent, then the family of lines 2ax + 3by + c = 0(a, b,c) are nonzero) is concurrent at (a) (2, 3) (b) ( (c) ) ( ) ( ) 205 1 2,1 -1 5 9 2 3, - 7 5 (d) 6, - 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Concurrency Of Three Lines Given A(0,0) and B(x,y) with x ∈ (0,1) and y>0. Let the slope of line AB be m1. Point C lies on line x = 1 such that the slope of BC is equal to m2 where0 < m2< m1 If the area ( 206 ) of triangle ABC can be expressed as m1- m2f(x) then the largest possible value of x is  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE LINES_Miscellaneous GEOMETRY_STRAIGHT If the vertices PandQ of a triangle PQR are given by (2, 5) and (4, - 11) , respectively, and the point R moves along the line N given by 9x + 7y + 4 = 0 , then the locus of the centroid of triangle PQR is a straight line parallel to PQ (b) QR (c) RP (d) N 207  Watch Free Video Solution on Doubtnut

  49. CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different Given A ≡ (1,1) and AB is any line through it cutting the x-axis at B. If AC is perpendicular to AB and meets the y-axis in C , then the equation of the locus of midpoint P of BC is x + y = 1 (b) x + y = 2x + y = 2xy (d) 2x + 2y = 1 208  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different The number of possible straight lines passing through (2, 3) and forming a triangle with the coordinate axes, whose area is 12 sq. units, is (a)one (b) two (c) three (d) four 209  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE LINES_Miscellaneous GEOMETRY_STRAIGHT In a triangle ABC , if A is (2, - 1), and7x - 10y + 1 = 0 and 3x - 2y + 5 = 0 are the equations of an altitude and an angle bisector, respectively, drawn from B , then the equation of BC is (a) a + y + 1 = 0 (b)5x + y + 17 = 0 (c)4x + 9y + 30 = 0 (d) x - 5y - 7 = 0 210  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different A triangle is formed by the lines x + y = 0,x - y = 0, and lx + my = 1. If landm vary subject to the condition l2+ m2= 1, then the locus of its circumcenter is (a) x2- y22= x2+ y2 (b) ( ) ( ) ( ) ( x2+ y22= 4x2y2 ) x2+ y22= x2- y2 211 (c) (d)

  50. x2- y22= x2+ y22 ( ) ( )  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different The line x + y = p meets the x- and y-axes at AandB , respectively. A triangle APQ is . inscribed in triangle OAB, O being the origin, with right angle at Q respectively, on OBandAB . If the area of triangle APQ is 3 P and Q lie, 212 8th of the are of triangle OAB, the AQ BQ is equal to (a)2(b) 2 3 (c) 1 3 (d)3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different A is a point on either of two lines y +√3|x| = 2 at a distance of 4√3 units from their point of intersection. The coordinates of the foot of perpendicular from A on the 213 ( ) ( ) 2 2 bisector of the angle between them are (a) - √3, 2 (b) (0, 0) (c) √3,2 (d) (0,4)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_STRAIGHT LINES_Equations Of Bisectors Of The Angle Between The Lines A pair of perpendicular straight lines is drawn through the origin forming with the line 2x + 3y = 6 an isosceles triangle right-angled at the origin. The equation to the line pair is 5x2- 24xy - 5y2= 05x2- 26xy - 5y2= 05x2+ 24xy - 5y2= 05x2+ 26xy - 5y2= 0 214  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Forms Of Line GEOMETRY_STRAIGHT LINES_Different If a pair of perpendicular straight lines drawn through the origin forms an isosceles triangle with the line 2x + 3y = 6 , then area of the triangle so formed is (a) 36 13 (b) 12 215 17 (c) (c) 13 5 (d) 17 14  Watch Free Video Solution on Doubtnut

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