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Cengage Maths Solutions Coordinate Geometry

Cengage Maths Solutions Class 12 Coordinate Geometry PDfs provides accurate solutions to practice before your final exam. Students can study this comprehensive Cengage Maths PDf Solutions on coordinate geometry 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 Coordinate Geometry

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  1. CENGAGE / G TEWANI MATHS SOLUTIONS CHAPTER COORDINATE GEOMETRY COORDINATE GEOMETRY ||  Download Doubtnut Today Ques No. Question CENGAGE_MATHS_COORDINATE GEOMETRY_Distance Formula GEOMETRY_COORDINATE In of Prove that , where is the middle point AB2+ AC2= 2(AO2+ BO2) ABC O 1 BC  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Distance Formula GEOMETRY_COORDINATE Find the coordinates of the circumcenter of the triangle whose vertices are and Find its radius also. (A(5, − 1),B( − 1,5), C(6,6). 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Distance Formula GEOMETRY_COORDINATE Two points Find the coordinates of and with another point form an equilateral triangle. O(0, 0) A(3, √3) P 3 P.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Distance Formula GEOMETRY_COORDINATE If (x1,y1)and(x2, y2), . OQ2cos∠Q1OQ2= x1x2 is the origin and if the coordinates of any two points respectively, prove that are O Q1andQ2 OQ1 4 + y1y2.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Distance Formula GEOMETRY_COORDINATE (3, 1), (6.5), ( , )

  2. Given that is a right angle and the area of and are three points such that the angle is 7, find the number of such points 5 P(3, 1),Q(6.5), R(x,y) PRQ RQP R.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE 6 Find the area of a triangle having vertices and A(3,2), B(11, 8), C(8,12).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE Prove that the area of the triangle whose vertices are is independent of (t + 3, t) and (t,t − 2),(t + 2, t + 2), 7 t.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE Find A(1, 1), B(7, − 3), C(12, 2), the area of the quadrilateral and D(7, 21). having vertices ABCD 8  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE For what value of are the points (k, 2 − 2k),( − k + 1, 2k)and( k 9 − 4 − k,6 − 2k) collinear?  Watch Free Video Solution on Doubtnut

  3. CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE If the vertices of a triangle have rational coordinates, then prove that the triangle cannot be equilateral. 10  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE If the coordinates of two points the coordinates of any point if and PA = PB. are (3, 4) and Area of , respectively, find is 10 sq. units. PAB (5, − 2) A B 11 P  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE Find the length of altitude through of the triangle A ≡ ( − 3, 0)B ≡ (4, − 1),C where ABC, A 12 ≡ (5, 2)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE The three points ( − 2,2)(9, − 2), 13 and( − 4, − 3) are the vertices of (a) an isosceles triangle (b) an equilateral triangle (c) a right-angled

  4. triangle (d) none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE The points vertices of a right angled triangle (c) vertices of an isosceles triangle (d) collinear are (a) vertices of an equilateral triangle (b) ( − a, − b),(a,b),(a2, ab) 14  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE Find the area of the pentagon whose vertices are A(1, 1), B(7, 21),C(7, − 3), 15 D(12, 2), and E(0, − 3)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE Let the number of position of and is a variable point on the lines with integral coordinates. B =6. IF |x| , then find A = (3,4) AB ≤ 4 B 16  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE If the points of θ and are collinear, then find the value (1,1):(0,sec2θ); (cosec2θ, 0) 17  Watch Free Video Solution on Doubtnut

  5. CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE Four points (AreaofDBC) and are given in such a way D(x,2x) A(6,3),B( − 3, 5),C(4, − 2) 18 1 that = . 2 (AreaofABC)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE Find the coordinates of the point which divides the line segments joining the points and in the ratio (i) internally and (ii) externally. (6,3) ( − 4, 5) 3:2 19  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE Given that such that and are two points and is a point on produced A(1, 1) B(2, − 3) D AB 20 Find the coordinates of AD = 3AB. D.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE Determine the ratio in which the line points (1,3) and (2, 7). divides the segment joining the 3x + y − 9 = 0 21  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE Prove that the points parallelogram. Is it a rectangle? and (1,2) are the vertices of a ( − 2, − 1),(1, 0), (4,3), 22  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE If divides P λ1;λ2, internally in the ratio then prove that and divides Q externally in the ratio OA λ1:λ2 OA 23 is the harmonic mean of OA and OP OQ.  Watch Free Video Solution on Doubtnut

  6. CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE If a vertex of a triangle is are and −2,3) , and the middle points of two sides passing through it then find the centroid and the incenter of the triangle. (1, 1) 24 (5,2),  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE 25 Find the orthocentre of the triangle whose vertices are and (0, 0),(3,0), (0, 4).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE If the circumcenter of an acute-angled triangle lies at the origin and the centroid is the middle point of the line joining the points find the orthocentre. and then (a2+ 1, a2+ 1) 26 (2a, − 2a),  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE If a vertex, the circumcenter, and the centroid of a triangle are (0, 0), (3,4), and (6, 8), respectively, then the triangle must be (a) a right-angled triangle (b) an equilateral triangle (c) an isosceles triangle (d) a right-angled isosceles triangle 27  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE Orthocenter and circumcenter of a the coordinates of the vertex point of BC. are then find the coordinates of the middle , respectively. If DeltaABC (a,b)and(c,d) 28 are (x1, y1), A

  7.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE If x12 + y12 = x22 + y22 = x32 and are three non-collinear points such that A(x1, y1),B(x2, y2), C(x3,y3) + y32, then prove that x1sin2A + x2sin2B 29 + x3sin2C = y1sin2A + y2sin2B + y3sin2C = 0.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE The vertices of a triangle are A( − 1, − 7), B(5, 1)andC(1, 4) 30 . If the internal angle bisector of meets the side in then find the length ∠B D, AC AD.

  8.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE If A(acosθ1,asinθ1), having vertices ABC B(acosθ2asinθ2), 31 andC(acosθ3, asinθ3) is equilateral, then prove that cosθ1+ cosθ2+ cosθ3= sinθ1 + sinθ2 + sinθ3 = 0.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE If the point (x, − 1), (3,y),( − 2,3), 32 and( − 3, − 2) taken in order are the vertices of a parallelogram, then find the values of xandy.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE If the midpoints of the sides of a triangle are find the coordinates of its vertices. then (2, 1),( − 1, − 3),and(4,5), 33  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE kc + la kd + lb The points and are (a) vertices of an ) ( (a,b), (c,d), , k + l k + l 34 equilateral triangle (b) vertices of an isosceles triangle (c) vertices of a right-angled triangle (d) collinear  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE If the middle points of the sides of a triangle are then find the centroid of the triangle. , ( − 2,3), (4, − 3),and(4, 5) 35  Watch Free Video Solution on Doubtnut

  9. CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE The line joining and is produced to the point Then prove that b:a. A(bcosαbsinα) B(acosβ,asinβ) so that and are in the ratio M(x, y) AM BM 36 β x + ytan(α + ) = 0. 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE is the centroid of a triangle and the coordinates of its any two vertices are and find the area of the triangle. (4, − 8) ( − 9, 7), If (1, 4) 37  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE In what ratio does the x=axis divide the line segment joining the points (5, 6)? and (2, − 3) 38  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE The vertices of If the circumcenter of a triangle are and A(x1,x1tanθ1),B(x2,x2tanθ2), coincides with the origin and cosθ1+ cosθ2 + cosθ3 C(x3, x3tanθ3). H(a,b) ABC 39 a is the orthocentre, show that = sinθ1+ sinθ2 + sinθ3 b  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_COORDINATE

  10. GEOMETRY_Coordinates Of Different Centers Of A Triangle If (x1+ 2)2+ (y1− 3)2 are the vertices of an equilateral triangle such that (xi,yi),i = 1, 2, 3, = (x2+ 2)2+ (y2− 3)2 40 = (x3+ 2)2+ (y3− 3)2, x1+ x2+ x3 then find the value of . y1+ y2+ y3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE The sum of the squares of the distances of a moving point from two fixed points (a,0) and is equal to a constant quantity ( − a, 0) 41 Find the equation to its locus. 2c2 .  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE Find the locus of a point, so that the join of angle at the moving point. and subtends a right ( − 5, 1) (3, 2) 42  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE A rod of length slides with its ends on two perpendicular lines. Find the locus of its midpoint. l 43  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE

  11. is a variable line sliding between the coordinate axes in such a way that lies on the x-axis and lies on the y-axis. If , and find the equation of the locus of PA = b,Pb = a AB = a + b, AB A 44 is a variable point on such that P. B P AB  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE Two points that ∠RPQ − ∠RQP are given. is a positive constant is a variable point on one side of the line Find the locus of the point 2α. such PandQ R PQ 45 R.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE If the coordinates of a variable point quantity, then find the locus of are where is a variable θ (acosθ,b sinθ), P 46 P.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE Find the locus of the point (t2− t + 1, t2+ t + 1),t ∈ R. 47  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE Find the locus of a point such that the sum of its distance from the points (0, 2) and is 6. (0, − 2) 48  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_COORDINATE

  12. GEOMETRY_Locus And Equation Of A Locus Find the locus of the foot of perpendicular from the point (2, 1) on the variable line passing through the point (0, 0). 49  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE A variable line through the point of the centroid of triangle meets the axes at is the origin). O . Find the locus P(2, 1) AandB 50 (where OAB  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE Find the locus of a point whose distance from (a, 0) is equal to its distance from the y- axis. 51  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE The coordinates of the point moves so that the locus of the point are (a,0) and when is constant, then find the equation to PA2− PB2= 2k2, k respectively. If a point ( − a, 0), AandB 52 P P.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE If A(cosα, sinα), 53 B(sinα, − cosα),C(1, 2) are the vertices of then as varies, find the locus of its centroid. α ABC,  Watch Free Video Solution on Doubtnut

  13. CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE Let moves on the line be the vertices of then find the locus of the vertex 2x + 3y = 1, If the centroid of the triangle A(2, − 3)andB( − 2, 1) ABC. 54 C.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE is a variable point whose locus is position of is the point of section of Find the locus of OP :PQ = 3:1. corresponding to a particular being the origin, such that OQ, O 2x + 3y + 4 = 0; Q 55 Q, P P.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE Find the locus of the middle point of the portion of the line which is intercepted between the axes, given that remains constant. xcosα + ysinα = p 56 p  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE Find the locus of the point of intersection of lines is a variable). xsinα − ycosα = b(α and xcosα + ysinα = a 57  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE A point moves such that the area of the triangle formed by it with the points (1, 5) and 58 .

  14. .units. Then, find the locus of the point. (3, − 7)is21sq  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE A variable line through point the circumcenter of triangle meets the axes at (where is the origin). OAB O . Find the locus of AandB P(2,1) 59  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE A straight line is drawn through respectively. If the rectangle to meet the axis of and at is completed, then find the locus of OACB , P(3,4) x y AandB 60 C.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Polar Coordinates GEOMETRY_COORDINATE 61 Convert into a polar equation. y = 10  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Polar Coordinates GEOMETRY_COORDINATE

  15. 62 Express the polar equation in rectangular coordinates. r = 2cosθ  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Polar Coordinates GEOMETRY_COORDINATE 63 Convert into a polar equation. x2− y2= 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Polar Coordinates GEOMETRY_COORDINATE 64 Convert into its equivalent Cartesian equation. rsinθ = rcosθ + 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Polar Coordinates GEOMETRY_COORDINATE Convert into its equivalent Cartesian equation. r = cosecθercosθ 65  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Polar Coordinates GEOMETRY_COORDINATE Find the maximum distance of any point on the curve origin. from the x2+ 2y2+ 2xy = 1 66  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Polar Coordinates GEOMETRY_COORDINATE 67 Convert into its equivalent Cartesian equation. r = 4tanθsecθ

  16.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Polar Coordinates GEOMETRY_COORDINATE Find the minimum distance of any point on the line origin using polar coordinates. from the 3x + 4y − 10 = 0 68  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Shifting Of Origin GEOMETRY_COORDINATE At what point should the origin be shifted if the coordinates of a point become (4,5) 69 ( − 3,9)?  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Shifting Of Origin GEOMETRY_COORDINATE Shift the origin to a suitable point so that the equation not contain a term in and the constant term. y will y2+ 4y + 8x − 2 = 0 70  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Shifting Of Origin GEOMETRY_COORDINATE The equation of curve referred to the new axes, axes retaining their directions, and origin is . Find the equation referred to the original axes. (4, 5) X2+ Y2= 36 71  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Shifting Of Origin GEOMETRY_COORDINATE

  17. Find the equation to which the equation x2+ 7xy − 2y2+ 17x − 26y 72 − 60 = 0 is transformed if the origin is shifted to the point parallel to the original axies. the axes remaining (2, − 3),  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Rotation Of Axis GEOMETRY_COORDINATE The 3x2+ 2xy + 3y2= 10. 450 curve Find its equation if the axes are rotated through an angle , the origin remaining unchanged. equation of a referred to a given system of axes is 73  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Rotation Of Axis GEOMETRY_COORDINATE Given the equation be rotated so that the term . Through what angle should the axes is removed from the transformed equation. xy 4x2+ 2√3xy + 2y2= 1 74  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Rotation Of Axis GEOMETRY_COORDINATE What does the equation 2x2+ 4xy − 5y2+ 20x − 22y 75 − 14 = 0 become when referred to the rectangular axes through the point new axes being inclined at an angle at , the ( − 2, − 3) with the old axes? 450  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Rotation Of Axis GEOMETRY_COORDINATE Determine so that the line passing through with the positive direction of the x-axis. 1350 makes an angle of (3,4)and(x,5) x 76  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Slope Of A Line GEOMETRY_COORDINATE If the point slope method. are collinear, then find the value of using (2,3), (1,1),and(x,3x) x, 77  Watch Free Video Solution on Doubtnut

  18. CENGAGE_MATHS_COORDINATE GEOMETRY_Slope Of A Line GEOMETRY_COORDINATE If three points are A( − 2,1)B(2,3), 78 andC( − 2, − 4) , then find the angle between ABandBC.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Slope Of A Line GEOMETRY_COORDINATE Angle of a line with the positive direction of the x-axis is . The line is rotated about some point on it in anticlockwise direction by angle Find the angle θ. θ 79 and its slope becomes 450 3.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Slope Of A Line GEOMETRY_COORDINATE Let be two given point. Find the slope of a line perpendicular to A(6,4)andB(2,12) 80 AB.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Angle Between Two Lines GEOMETRY_COORDINATE If line value of is parallel to the line then find the 3x − ay − 1 = 0 (a + 2)x − y + 3 = 0 81 a.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Angle Between Two Lines GEOMETRY_COORDINATE (2, − 1) (6, 5)

  19. If perpendicular from are two points, then find the ratio in which the food of the to divides it. (4,1) AB 82 A(2, − 1)andB(6, 5)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Angle Between Two Lines GEOMETRY_COORDINATE 83 Find the orthocentre of with vertices and ΔABC A(1, 0), B( − 2,1), C(5, 2)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Angle Between Two Lines GEOMETRY_COORDINATE If (b2− b1)(b3− b1) + (a2− a1)(a3− a1) = 0 84 , then prove that the circumcenter of the triangle having vertices and (a1,b1), (a2,b2) a2+a3 b2+b3 is ( ) (a3,b3) , 2 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Angle Between Two Lines GEOMETRY_COORDINATE The line joining the points direction of the x-axis. Then find the values of makes an obtuse angle with the positive (x,2x)and(3, 5) 85 x.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Angle Between Two Lines GEOMETRY_COORDINATE If the line passing through find the value of is parallel to the line then (4, 3)and(2,k) y = 2x + 3, 86 k.

  20.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Angle Between Two Lines GEOMETRY_COORDINATE 87 Find the orthocentre of with vertices and A(1, 0),B( − 2, 1), C(5,2) ABC  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE If the points y2− y3 and are collinear show that (x3,y3) (x1,y1), (x2, y2), y3− y1 y1− y2 + + 88 x1x2 x2x3 x3x1 = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE Let AP2+ PC2= PB2+ PD2 be a rectangle and ABCD be any point in its plane. Show that P 89 .  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE If (bc,p), (ca, q), are the terms, respectively, of an are collinear. , show that the points a, b,c pth,qth, rth HP 90 and (ab, r)  Watch Free Video Solution on Doubtnut

  21. CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE Prove that the circumcenter, orthocentre, incenter, and centroid of the triangle formed by the points actually finding any of them. 91 and are collinear, without A( − 1,11),B( − 9, − 8), C(15, − 2)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE A rod of length slides in a vertical plane, its ends touching the coordinate axes. Prove that the locus of the foot of the perpendicular from the origin to the rod is k (x2+ y2) . 92 3= k2x2y2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE and any point the side of are two coordinate axes. On On an equilateral triangle is described, its vertex away from . Then prove that the locus of PQ O is taken a fixed point and on being on P(0,c) OX OY OY 93 PQ, OX Q. R is y = √3x − ⋅ R  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Miscellaneous GEOMETRY_COORDINATE If rectangular axes with the same origin and it independent of , becomes xandy and are the coordinates of the same point referred to two sets of ux + vy, (x,y) (x,y) 94 where u2+ v2= U2+ V2 and u are v show that V X + UY , .  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Distance Formula GEOMETRY_COORDINATE is an isosceles triangle. If the coordinates of the base are and B(1,3) ABC 1 5 95 , the coordinates of vertex C( − 2,7) can be (b) (d) ( − , 5) ( ,6) 6 (1, 6) A 2 none of these  Watch Free Video Solution on Doubtnut

  22. CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE If two vertices of a triangle are (1,3) and (4,-1) and the area of triangle is 5 sq. units, then the angle at the third vertex lies in : 96  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE Which of the following sets of points form 3 an 2 equilateral triangle? 3 4 4 2 97 (d) (1,0), (4,0),(7, − 1) (0,0),( ),( ) ( ,),(0, ),(1, 1) , , 2 3 3 2 3 3 None of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Miscellaneous GEOMETRY_COORDINATE A particle moves from the point travel the upper-half plane plane axis, if the sum of the squares of the travel times of the upper- and lower-half planes is minimum, are (1, 0) (b) (2, 0) (c) (4, 0) (d) (5, 0) to the point at the speed of . The particle and the lower-half 1m/s can A(0,4) 10, − 4) p P {(x,y) ∣ y ≥ } 98 at the speed of 2 m/s. The coordinates of a point on the x- {(x, y) ∣ y ≤ 0}  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE If |x1y11x2y21x3y31| 99 = |a1b11a2b21a3b31| then the two triangles with vertices and (x1, y1),(x2, y2),(x3,y3) are equal to area (b) similar congruent (d) none of these (a1,b1), (a2,b2),(a3, b3)  Watch Free Video Solution on Doubtnut

  23. CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE is a square and respectively. Then the ratio of the area of the square to that of triangle (b) 2:1 (c) 8:3 (d) 7:3 are the middle points of the sides , M, N OPQR PQandQR 100 is 4:1 OMN  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Polar Coordinates GEOMETRY_COORDINATE A straight line passing through given that the distance of this straight line from the origin 50 meets the coordinate axes at P(3,1) . It is AandB is maximum. The area of 100 O 101 25 20 .units .units .units .units triangle is equal to (b) (d) OAB sq sq sq sq 3 3 3 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Polar Coordinates GEOMETRY_COORDINATE Let such that Let be a variable point A ≡ (3, − 4), B ≡ (1,2). P ≡ (2k − 1,2k + 1) 102 is the minimum. Then is 7/9 (b) 0 (c) 7/8 (d) none of these PA + PB k  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Polar Coordinates GEOMETRY_COORDINATE If and are the coordinates of three points, then the value C(p,0) A(1,p2),B(0,1) 1 1 1 103 of for which the area of triangle p is the minimum is (b) (d) − ABC √3 √3 √2 none of these  Watch Free Video Solution on Doubtnut

  24. CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE If the point (x1+ t(x2− x1), y1 104 + t(y2− y1)) divides the join of and internally, then (b) `01 t=1` (x1,y1) (x2, y2) t < 0 (d)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Distance Formula GEOMETRY_COORDINATE and AP = PQ = QB are points on the line joining . Then, the distance of the midpoint of and such that A( − 2,5) B(3,1) P Q from the origin is (a) 3 PQ 105 √37 (b) (b) (d) 4 3.5 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE If two vertices of a triangle are origin, and the centroid on the line (d) none of these 8,14) (12, 21) and the orthocentre lies at the , then the third vertex lies at ( − 2, 3) (5, − 1) (b) 106 x + y = 7 (7, 4)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE 1 1 1 The vertices of a triangle are and (pq, ),(pq)), (qr, ), (rq, ), pq qr rp 107 where coordinates of its centroid are and are the roots of the equation r (1,2) . The y3− 3y2+ 6y + 1 = 0 p,q (b) (d) 2, − 1) (1, − 1) 2,3)  Watch Free Video Solution on Doubtnut

  25. CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE If the vertices of a triangle are , ( , and , then the (√5, 0) √3, √2) (2, 1) 108 orthocentre of the triangle is of these (b) (d) none (√5,0) (0,0) (√5 + √3 + 2, √2 + 1)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE In if the orthocentre is and the circumcenter is (0, 0), then centroid of (1,2) ABC, 109 1 2 1 2 2 is (b) (d) none of these ( , 2 ) ( , 3 ) ( , 1) 3 ABC) 3 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE 3 7 A triangle with vertices and has A( − 1,0),B( − 2, ), C( − 3, − ) ABC 4 6 110 its orthocentre at 1,3) ( − 1,2) Then, the orthocentre of triangle (d) none of these will be (b) ( − 3, − 2) H. BCH  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE 1 2 If a triangle circumcenter and orthocentre ≡ ( − ), ABC,A ≡ (1, 10), , 3 3 11 4 111 , then the coordinates of the midpoint of the side opposite to are A ≡ ( ) , 4 3 11

  26. 11 (b) ) (d) (1, − (1,5) (1, − 3) (1,6) 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE π In the coordinates of are and the middle ABC, (0, 0),AB = 2,∠ABC = , B 3 1 √3 ( , (b) ) point of has coordinates The centroid o the triangle is (2, 0). BC 112 2 2 5 1 √3 1 (4 + (d) none of these ) ( ) , , 3 3 3 √3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE ab If the origin is shifted to the point without rotation, then the equation ( , 0) a − b becomes (A) (a − b)(x2+ y2) − 2abx = 0 (a − b)(x2+ y2) − (a + b)xy 113 + abx = a2 (a + b)(x2+ y2) = 2ab (B) (C) (D) (x2+ y2) = (a2+ b2) (a − b)2(x2+ y2) = a2b2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Distance Formula GEOMETRY_COORDINATE A light ray emerging from the point source placed at is reflected at a point P(2,3) Q (5, 10)

  27. 114 on the y-axis. It then passes through the point (b) (d) none of these (0,3) (0,2) (0, 5) The coordinates of R(5, 10). are Q  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE Point quadrilateral (d) none of these from parallelogram rhombus cyclic P(p,0), Q(q,0), R(0,p), S(0,q) 115  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Miscellaneous GEOMETRY_COORDINATE A rectangular billiard table has vertices at A small billiard ball starts at the table, bounces to the right side of the table, and then comes to rest at The coordinate of the point where it hits the right side is (a) y − 4 and P(0,0), Q(0,7),R(10,7), , moves in a straight line to the top of M(3,4) S(10,0). 116 . N(7, 1) 3.9 (b) (c) (d) 3.7 3.8  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE If one side of a rhombus has endpoints (4, 5) and (1, 1), then the maximum area of the rhombus is 50 sq. units (b) 25 sq. units 30 sq. units (d) 20 sq. units 117  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Miscellaneous GEOMETRY_COORDINATE A rectangle A ≡ (0, 0),B ≡ (4,0),C where ABCD, ≡ (4, 2)D ≡ (0, 2) 118 f1(x, y)− − → , f2(x,y)− − − − − − → undergoes x + 3y,y f3(x,y)− − − − − − − → the following transformations successively: y,x The final figure will be square (x − y)/2, (x + y)/2) (b) a rhombus a rectangle (d) a parallelogram  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE If a straight line through the origin bisects the line passing through the given points and (a cosα, asinα) (acosβ, asinβ),π then the lines (a)are perpendicular (b)are 119 parallel (c)have an angle between them of (d)none of these 4

  28.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Miscellaneous GEOMETRY_COORDINATE Let are in proper fraction. Let , be the points on the number line such that is the origin, and the common ratio of the be the middle point of the line segment M, Ar,r = 1, 2, 3, GP, OA1, OA2,OA3. GP where be a positive Then the ArAr+1. O 120 ∞ value of is equal to ∑ OMr r=1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE The vertices of a parallelogram If a line passing through the origin divides the parallelogram into two D(3,16). are and A(3,1), B(13, 6),C(13,21), ABCD 121 11 11 25 13 congruent parts, then the slope of the line is (b) (c) (d) 12 8 8 8  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE Point and A are in the first quadrant; point B is the origin. If the slope of O is 1, −1 OA −1 −1 the slope of OB is 7, and then the slope of is (b) (c) OA = OB, AB 122 5 4 3 −1 (d) 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE is a square and OPQR are the midpoints of the sides and , M, N PQ QR :6,

  29. 123 respectively. If the ratio of the area of the square to that of triangle λ is λ:6, OMN then is equal to 2 (b) 4 (c) 2 (d) 16 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE If 4 ∑ (ξ2 + yi2) ≤ 2x1x3+ 2x2x4 i− 1 124 + 2y2y3+ 2y1y4, the points (x1,y1), (x2, y2),(x3,y3), (x4,y4) are the vertices of a rectangle collinear the vertices of a trapezium none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE The vertices y − aξs, and A of square lie on the positive sides of is the point (d) 17,5) (17,12) and x − D ABCD respectively. If the vertex are (b) (15, 3) B (14,16) , then the coordinates of 125 (12,17) C vertex  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE y x Through the point , where the straight line is drawn P(α,β) αβ > 0, + = 1 then the least value of a b 126 so as to form a triangle of area with the axes. If is (b) (c) (d) none αβ 2αβ 3αβ ab > 0, S S  Watch Free Video Solution on Doubtnut

  30. CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE The locus of the moving point whose coordinates are given by (et+ e−t,et− e− t) x2− y2= 2 127 where is a parameter, is t (b) (d) x + y = 2 x2− y2= 4 xy = 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE The locus of a point reprersented by t + 1 a ( ),y x = 2 t t − 1 a 128 ( ) = 2 1 , where x − y = a is (b) (d) x2+ y2= a2 x2− y2= a2x + y = a t ∈ R − {0},  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Miscellaneous GEOMETRY_COORDINATE The maximum area of the triangle whose sides a,b and c satisfy 0 ≤ a ≤ 1, 1 ≤ b ≤ 2 and 2 129 ≤ c ≤ 3 is  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE If remaining vertex can be (a) are the vertices of a parallelogram, then the (b) (c) (d) (0, − 1) 7, 9) ( − 1,0) ( − 6, − 4),(3, 5),( − 2,1) 130 ( − 11, − 8)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE Let 0 ≡ (0,0),A ≡ (0,4),B 131 ≡ (6, 0). P Let be a moving point such that the area of triangle triangle . The locus of will be a straight line whose equation can be POB P is two times the area of POA  Watch Free Video Solution on Doubtnut

  31. CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE .units, If third vertex lies on x + 5y + 12 = 0 and are two vertices of a triangle of area (b) y = x 5x + y + 12 = 0 x + 5y − 4 = 0 then its (d) ) − 4,0) (1, − 1) 4sq 132  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE The area of triangle those of are (b) (5, 3) is The coordinates of vertex lies on the line (d) (7, 5) are A and 20cm2 −5, 0) ABC . 133 are The vertex . The coordinates of (3,0). ( − 3, − 5) ( − 5, − 7) x − y = 2 B C C  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE If (acosθ1, asinθ1), 134 (acosθ2, asinθ2) (acosθ3, asinθ3) and circle, then (a) tanθ1+ tanθ2+ tanθ3= 0 represent the vertices of an equilateral triangle inscribed in a (b) (d) cotθ1+ cotθ2+ cotθ3= 0 (c) cosθ1+ cosθ2+ cosθ3= 0 sinθ1+ sinθ2+ sinθ3= 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE The points and are the vertices of a right- A(0, 0),B(cosα, sinα) C(cosβ,sinβ) α − β 1 α − β 1 angled triangle if (a) (b) (c) sin( ) = cos( ) = − 135 2 2 √2 √2 α − β 1 α − β 1 (d) cos( ) = sin( ) = − 2 2 √2 √2

  32.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE The ends of a diagonal of a square are and Another vertex of the (2, − 3) ( − 1, 1). 5 136 3 5 5 1 1 square can be (b) ) (d) none of these ) ( − ( ) ( , − , , 2 2 2 2 2 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE If all the vertices of a triangle have integral coordinates, then the triangle may be (a) right-angle(b) equilateral (c) isosceles(d) none of these 137  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Miscellaneous GEOMETRY_COORDINATE In a ABC,A ≡ (α,β), B ≡ (1, 2), C ≡ (2, 3), 138 point triangle is such that the possible coordinates of lies on the line where [ ] are integers, and the area of the y = 2x + 3, α,β A where . denotes the greatest integer function. Then can be (a) (b) A ( − 7, − 11) [S] = 2 S (c) (d) ( − 6, − 9) (2,7) (3,9)

  33.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE In an acute triangle are 12 (d) −12 , if the coordinates of orthocentre , and of circumcenter are But no common value of is possible. are , of centroid (4,b) ABC H 139 , then cannot be (b) (c) b (b, 2b − 8) ( − 4, 8) 4 8 G S b  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Miscellaneous GEOMETRY_COORDINATE Consider the points points  Watch Free Video Solution on Doubtnut , and in the plane. Suppose that 140 O(0,0),A(0, 1) B(1, 1) x,y and are chosen such that `0 C(x,1) D(1, y) CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE Statement 1 : If the vertices of a triangle are having rational coordinates, then its centroid, circumcenter, and orthocentre are rational. Statement 2 : In any triangle, orthocentre, centroid,and circumcenter are collinear, and the centroid divides the line joining the orthocentre and circumcenter in the ratio 2:1. 141  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE Statement 1 : The area of the triangle formed by the points A(1000,1002), B(1001,1004), 142 C(1002,1003) is the same as the area formed by the point 2 : The area of the triangle is constant with respect to the translation of axes. Statement A′(0,0),B′(1,2), C′(2,1)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Distance Formula GEOMETRY_COORDINATE Statement 1 :If the lines at and the y-axis at Statement 2 : Since concyclic. and cut the x-axis are concyclic. A, B,C,D 2x + 3y + 19 = 0 C, D, OAxOB = OCxOD, 9x + 6y − 17 = 0 A, B, C,D then the points, where A,B 143 is the origin, are O  Watch Free Video Solution on Doubtnut

  34. CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE Statement 1 : Let the vertices of a . Then the coordinates of the circumcenter are right-angled triangle, the midpoint of the hypotenuse is the circumcenter of the triangle. be and A( − 5, − 2), B(7,6), ABC Statement 2 : In a C(5, − 4) (1, 2). 144  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Miscellaneous GEOMETRY_COORDINATE Each equation contains statements given in two columns which have to be matched. Statements (a,b,c,d) in column I have to be matched with Statements (p, q, r, s) in a→ p ,a→ s ,b→ 4x4 (x2+ xy − x)x(x − y) = 0, q ,b→ r , c→ p ,c→ d→ s column II. If the correct match are correctly bubbled represented by equation and , then the q , matrix should be as follows: Figure Consider the lines forming a triangle. Then 145 1 1 match the following: Column I|Column II Orthocenter of triangle |p. ( ) , 6 2 1 1 1 1 Circumcenter|q. Centroid|r. ) Incenter|s. ) (1(2 + 2√2), (0, ( , 2 ) 2 2 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE d2y y = aemx+ be−mx, dx2 146 = m2yisequal → m2(aemx− bemx)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Section Formula GEOMETRY_COORDINATE

  35. A point integral value of for which the area of , is equal to 2 units in magnitude is___ divides the join of and in the ratio is B (1,5) . Then the is (7, − 2) P( − 5,1) Q(3, 5) k:1 A 147 where and ABC, k C  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE The distance between the circumcenter and the orthocentre of the triangle whose 2 148 vertices are and is Then the value of is_________ (0,0), (6,8), ( − 4,3) L. L √5  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Distance Formula GEOMETRY_COORDINATE A man starts from the point axis at R(α,0) and reaches the point is minimum. Then touching the x- _________ P( − 3, 4) Q(0,1) 149 such that PR + RQ 5|α| =  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Distance Formula GEOMETRY_COORDINATE Let A(0, 1)B(1,1),C(1, − 1), 150 D( − 1, 0) be four points. If is_____, where [.] represents greatest integer. [D] is any other points, then when PA + PB + PC + PD ≥ D, P  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Distance Formula GEOMETRY_COORDINATE

  36. The vertices of a triangle are A( − 1, − 7), B(5, 1)andC(1, 4) 151 . If the internal angle bisector of meets the side in then find the length ∠B D, AC AD.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE If the area of the triangle formed by the points is then 2q (a + b, a − b), (3b − a, b + 3a), and vertices (2a,b)(a + b, 2b + a), (2b, 2a) .units, 152 the area of the triangle whose will be_____ are and (3a − b, 3b − a)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE The area of a triangle is 5. Two of its vertices are vertex is on C y = x + 3. and . The third A(2, 1) B(3, − 2) 153 Find C.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE Two vertices of a triangle are is the origin, find the coordinates of the third point. and If the orthocentre of the triangle ( − 2,3) (5, − 1) 154  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE The vertices of a Then triangle are , , [at1t2,a(t1+ t2)] [at2t3,a(t2+ t3)] the orthocenter (b) ( − a, a(t1+ t2+ t3) + at1t2t3) (a,a(t1+ t2+ t3) − at1t2t3) of the triangle is (a) (c) [at3t1,a(t3+ t1)] ( − a, a(t1+ t2+ t3) − at1t2t3) (a, a(t1+ t2+ t3) + at1t2t3) 155 (d)  Watch Free Video Solution on Doubtnut

  37. CENGAGE_MATHS_COORDINATE GEOMETRY_Area Of A Triangle/Polygon GEOMETRY_COORDINATE The coordinates of any point |x + y − 2| are , respectively, and to that of PBC is is A, B,C (6,3),( − 3,5), (4, − 2) P . Show that the ratio of the area of (x,y) ABC 156 . 7  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Locus And Equation Of A Locus GEOMETRY_COORDINATE A line cuts the x-axis at drawn perpendicular to AB cutting the x-axis in P and the y-axis in Q. If AQ and BP intersect at R, find the locus of R and the y-axis at A(7,0) A variable line PQ is B(0, − 5) 157  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Distance Formula GEOMETRY_COORDINATE The point (4, 1) undergoes the following three transformations successively: (a) Reflection about the line y = x (b) Translation through a distance 2 units along the positive direction of the x-axis. (c) Rotation through an angle 158 π about the origin in the 4 anti clockwise direction. The final position of the point is given by the co-ordinates.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Miscellaneous GEOMETRY_COORDINATE If points circle (d) are the vertices of a triangle. as well as (x1,y1), (x2, y2), are in (x3,y3). with the same common ratio, then the lie on a straight line lie on an ellipse lie on a x1,x2, x3 y1,y2,y3 GP 159 and  Watch Free Video Solution on Doubtnut

  38. CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE The incenter of the triangle with vertices and is (a) (1, √3),(0,0), (2,0) 160 √3 2 1 2 √3 1 (1, ) ( , ) (b) (c) (d) ( , 3 ) (1, ) 2 3 2 √3 √3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Miscellaneous GEOMETRY_COORDINATE 161 Let `0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Miscellaneous GEOMETRY_COORDINATE Number of points with integral co-ordinates that lie inside a triangle whose co- ordinates are (0, 0), (0, 21) and (21,0). 162  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Coordinates Of Different Centers Of A Triangle GEOMETRY_COORDINATE 5 The orthocentre of the triangle with vertices and is (a) (3, ) (0,0), (3, 4), (4,0) 4 163 3 (b) (c) (d) (3, ) (3,12) (3,9) 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Distance Formula GEOMETRY_COORDINATE (0,0), (3,4), (6,0)

  39. Let inside the triangle and is such that the triangles OPQ 4 be the vertices of triangle Q(6,0) . The point are of equal 4 2 O(0,0), P(3,4), OPQ R OPR,PQR,OQR 164 2 4 area. The coordinates of are R (b) (d) ( ,3) (3, ) (3, ) ( , 3 ) 3 3 3 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Distance Formula GEOMETRY_COORDINATE Consider three points and R , , P = ( − sin(β − α), − cosβ) Q = (cos(β − α), sinβ) 165 = ((cos(β − α + θ),sin(β − θ)) 0 < α,β,θ <π , where Then 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Distance Formula GEOMETRY_COORDINATE 8 The points , and are the vertices of an obtuse-angled (0, ),(1,3) (82,30) 166 3 triangle an acute-angled triangle a right-angled triangle none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_Distance Formula GEOMETRY_COORDINATE If PQRS, a = 1 or b = − 1 and are the vertices of a parallelogram (c) a = 3,b = 4 a = 2, b = 3 P(1, 2)Q(4,6),R(5,7), S(a,b) 167 then (a) (b) (d) a = 2,b = 4  Watch Free Video Solution on Doubtnut

  40. CENGAGE_MATHS_COORDINATE GEOMETRY_Miscellaneous GEOMETRY_COORDINATE If the vertices P, Q, R of a triangle PQR are rational points, which of the following points of the triangle POR is (are) always rational point(s) ? 168  Watch Free Video Solution on Doubtnut  Download Doubtnut to Ask Any Math Question By just a click  Get A Video Solution For Free in Seconds  Doubtnut Has More Than 1 Lakh Video Solutions  Free Video Solutions of NCERT, RD Sharma, RS Aggarwal, Cengage (G.Tewani), Resonance DPP, Allen, Bansal, FIITJEE, Akash, Narayana, VidyaMandir  Download Doubtnut Today

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