Cengage Maths Solutions Hyperbola - Coordinate Geometry

1. CENGAGE / G TEWANI MATHS SOLUTIONS CHAPTER HYPERBOLA || COORDINATE GEOMETRY  Download Doubtnut Today Ques No. Question CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola Find the equation of hyperbola : whose axes are coordinate axes and the distances of one of its vertices from the foci are 3 and 1 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola Find the equation of hyperbola : Whose center is (1,0), focus is (6,0) and the transverse axis is 6 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola Find the equation of hyperbola : Whose center is (3, 2), one focus is (5, 2) and one vertex is (4, 2) 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola Find the equation of hyperbola : Whose center is 5 one vertex is ( − 3,2), ( − 3,4), 4 and eccentricity is . 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola 5 Find the equation of hyperbola : Whose foci are (4, 2) and (8, 2) and accentricity is 2.  Watch Free Video Solution on Doubtnut

2. CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola If the base of a triangle and the ratio of tangent of half of base angles are given, then identify the locus of the opposite vertex. 6  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola Two circles are given such that they neither intersect nor touch. Then identify the locus of the center of variable circle which touches both the circles externally. 7  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola Two rods are rotating about two fixed points in opposite directions. If they start from their position of coincidence and one rotates at the rate double that of the other, then find the locus of point of the intersection of the two rods. 8  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola 9 Find the vertices of the hyperbola 9x2− 16y2− 36x + 96y − 252 = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola Find the coordinates of the foci, the eocentricity, the latus rectum, and the equations of directrices for the hyperbola 9x2− 16y2− 72x + 96y − 144 = 0 10  Watch Free Video Solution on Doubtnut

3. CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola y2 y2 x2 x2 1 If the foci of the ellipse and the hyperbola + = 1 − = 11 b2 16 144 81 25 coincide, then find the value of b2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola y2 y2 x2 x2 If hyperbola passes through the focus of ellipse = 1 , − + = 1 12 b2 a2 a2 b2 then find the eccentricity of hyperbola.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola An ellipse and a hyperbola have their principal axes along the coordinate axes and have a common foci separated by distance axes is equal to 4. If the ratio of their eccentricities is 3/7 , find the equation of these curves. The difference of their focal semi- 2√3. 13  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola Find the eccentricity of the hyperbola given by equations et+ e−1 et− e−1 14 x = andy = , t ∈ R 3 2 .  Watch Free Video Solution on Doubtnut

4. CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola y2 x2 If is a double ordinate of the hyperbola PQ such that = 1 is an − OPQ 15 a2 b2 equilateral triangle, eccentricity of the hyperbola. e being the center of the hyperbola, then find the range of the O  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola If the latus rectum subtends a right angle at the center of the hyperbola x2 y2 16 , then find its eccentricity. = 1 − a2 b2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola If the latus rectum of a hyperbola forms an equilateral triangle with the vertex at the center of the hyperbola ,then find the eccentricity of the hyperbola. 17  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola The distance between two directrices of a rectangular hyperbola is 10 units. Find the distance between its foci. 18  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola

5. An ellipse and a hyperbola are confocal (have the same focus) and the conjugate axisAn ellipse and a hyperbola are confocal (have the same focus) and the conjugate axis of the hyperbola is equal to the minor axis of the ellipse. If eccentricities of the ellipse and the hyperbola, respectively, then prove that 1 1 are the e1ande2 19 . + = 2 e12 e22  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola Two straight lines pass through the fixed points products is Show that the locus of the points of intersection of the lines is a hyperbola. and have slopes whose ( ± a,0) 20 p > 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: Find the lengths of the transvers and the conjugate axis, eccentricity, the coordinates of foci, vertices, the lengths of latus racta, and the equations of the directrices of the following hyperbola: 16x2− 9y2= − 144. 21  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: If show that the locus of a points hyperbola. Find its eccentricity. are two straight lines which bisect one another at right angles, which moves so that P AOBandCOD 22 is a PAxPB = PCxPD  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: If show that are the foci, SP × S′P = (CP)2 is the center, and is a point on a rectangular hyperbola, SandS' C P 23 .  Watch Free Video Solution on Doubtnut

6. CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: Find the equation of the hyperbola whose foci are (8, 3) and (0, 3) and eccentricity is 4 24 . 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: The equation of one of the directrices of a hyperboda is corresponding focus is (1, 2) and the coordinates of the center and the second focus. the 2x + y = 1, . Find the equation of the hyperbola and 25 e = √3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: and are fixed straight lines, on is of constant area. OMPN is any point and respectively. Find the locus of and are the if the P OA OB P PM PN 26 perpendiculars from quadrilateral OAandOB, P  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: Find the coordinates of the foci and the centre of the hyperbola (3x − 4y − 12)2 ( ) 100 27 (4x + 3y − 12)2 − ( ) = 1 225  Watch Free Video Solution on Doubtnut

7. CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: 28 4(2y − x − 3)2− 9(2x + y − 1)2= 80 Find the eccentricity of the conic  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: For all real values of which of the following certain hyperbolas? (a) (c) (d) 9x2− 4y2= 36 the straight line is a tangent to 4x2+ 9y2= 36 y = mx +√9m2− 4 9x2+ 4y2= 36 m, (b) 29 4x2− 9y2= 36  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: Find the value of x2 y2 for which m is tangent to the hyperbola y = mx + 6 30 − = 1 100 49  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: Find the equation of tangents to the curve 4y = 5x + 7. which are parallel to 4x2− 9y2= 1 31  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola:

8. 32 Find the point on the hyperbola it. where the line touches x2− 9y2= 9 5x + 12y = 9  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: 33 Find the equation of tangent to the conic at x2− y2− 8x + 2y + 11 = 0 (2,1)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: Find the equations of the tangents to the hyperbola from (3, 2). that are drawn x2− 9y2= 9 34  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 If it is possible to draw the tangent to the hyperbola having slope 2, − = 1 35 a2 b2 then find its range of eccentricity.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: 36 Find the equation of the common tangent to the curves and xy=-1. y2= 8x  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Hyperbola:

9. Definition 2 y2 x2 Find the equations to the common tangents to the two hyperbolas − = 1 a2 b2 37 y2 x2 and − = 1 a2 b2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 is a point on the hyperbola is the foot of the perpendicular from − = 1,N P a2 b2 38 on the transverse axis. The tangent to the hyperbola at at If is the center of the hyperbola, then find the value of T. O meets the transvers axis OTxON. P P  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: On which curve does the perpendicular tangents drawn to the hyperbola x2 y2 39 intersect? = 1 − 25 16  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 Tangents are drawn to the ellipse at two points whose eccentric = 1 + 40 a2 b2 angles are and The coordinates of their point of intersection are α − β α + β  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: Tangents are drawn from the points on a tangent of the hyperbola the parabola If all the chords of contact pass through a fixed point prove that the locus of the point for different tangents on the hyperbola is an ellipse. Q to x2− y2= a2 41 y2= 4ax. Q,  Watch Free Video Solution on Doubtnut

10. CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: How many real tangents can be drawn from the point (4, 3) to the hyperbola x2 y2 42 Find the equation of these tangents and the angle between them. − = 1? 16 9  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: Which of the following can be slope of tangent to the hyperbola −3 2 1 (b) 4x2− y2= 4? 43 (c) 2 (d) −3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: 5 Tangents are drawn to the hyperbola from the point Find ). 3x2− 2y2= 25 (0, 44 2 their equations.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 From the center of hyperbola , perpendicular is drawn on any − = 1 C CN a2 b2 45 tangent to it at the point so that the area of in the first quadrant. Find the value of P(asecθ,btanθ) θ is maximum. CPN

11.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: 46 A common tangent to and , is 9x2− 16y2= 144 x2+ y2= 9  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: Find the equation of tangents to the curve 4y = 5x + 7. which are parallel to 4x2− 9y2= 1 47  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: The locus a point moving under the condition that the line x2 y2 is a P(α,β) y = αx + β 48 tangent to the hyperbola is − = 1 a2 b2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: A point parabola moves such that the chord of contact of the pair of tangents from touches the rectangular hyperbola y2= 4ax x2 y2 on the P P Show that the x2− y2= c2 . 49 locus of is the ellipse P + = 1. c2 (2a)2  Watch Free Video Solution on Doubtnut

12. CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 is the ordinate of any point on the hyperbola and is its − = 1 ∀' PN P 50 a2 b2 transvers axis. If perpendicular to divides Q A′P. in the ratio then prove that is a2:b2, AP NQ  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 is the center of the hyperbola The tangent at any point on this − = 1 C P a2 bx − ay = 0 . CR = a2+ b2 b2 51 hyperbola meet the straight lines and at points , bx + ay = 0 QandR respectively. Then prove that CQ .  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: 52 Find the equation of normal to the hyperbola at point (4, 1). x2− 9y2= 7  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: 1 53 Find the equation of normal to the hyperbola having slope 3x2− y2= 1 . 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola:

13. y2 x2 If the normal at on the hyperbola P(θ) meets the transvers axis at = 1 − a2 2a2 54 . ′G = a2(e4sec2θ − 1) then prove that of the hyperbola. , where are the vertices G, AandA' AG A  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 Normal are drawn to the hyperbola at point meeting the − = 1 θ1andthη2 a2 b2 π conjugate axis at respectively. If prove that G1andG2, θ1+ θ2= , 55 2 a2e4 . , where is the center of the hyperbola and is the e CG2= CG1 C e2− 1 eccentricity.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: Normal is drawn at one of the extremities of the latus rectum of the hyperbola x2 y2 56 which meets the axes at points = 1 . Then find the area of − AandB a2 b2 OAB(O triangle being the origin).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And A ray emanating from the point (5, 0) is INCIDENT on the hyperbola at the point with abscissa 8. Find the equation of the reflected ray after the first reflection if point lies in the first quadrant. P 57 9x2− 16y2= 144 P  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And If and be two coordinate of the ends of a focal y2 tan( )tan( θ 2 (asecθ, btanθ) (asecϕ,btanϕ) x2 58 ϕ chord passing through of then equals to ) (ae,0) − = 1 a2 b2 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And 2 2

14. 59 y2 x2 If the tangents to the parabola intersect the hyperbola at y2= 4ax − = 1 a2 b2 AandB. , then find the locus of the point of intersection of the tangents at AandB  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And The locus ofthe midde points ofchords of hyperbola parallel to is y = 2x 3x2− 2y2+ 4x − 6y = 0 60  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And Find the condition on x2 y2 for which two distinct chords of the hyperbola aandb 61 passing through = 1 are bisected by the line (a, b) . − x + y = b 2a2 2b2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And y2 x2 If any line perpendicular to the transverse axis cuts the hyperbola and − = 1 a2 b2 62 y2 x2 the conjugate hyperbola at points , respectively, then − = − 1 PandQ a2 b2 prove that normal at meet on the x-axis. PandQ  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And 2 2

15. y2 x2 A normal to the hyperbola meets the axes at and lines − = 1 63 MandN a2 b2 are drawn perpendicular to the axes meeting at of is the hyperbola P a2x2− b2y2= (a2+ b2). and Prove that the locus P. MP NP  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And Find the equation of the chord of the hyperbola bisected at the point (5, 3). which is 25x2− 16y2= 400 64  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And Find the equation of the locus of the middle points of the chords of the hyperbola each of which makes an angle of 2x2− 3y2= 1, 65 with the x-axis. 450  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And Prove that the locus of the point of intersection of the tangents at the ends of the normal chords of the hyperbola x2− y2= a2 66 is a2(y2− x2) = 4x2y2 .  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And If x2 , then the chord joining the points α + β = 3π y2 and for the hyperbola β α 67 passes through which of the following points? Focus (b) Center One = 1 − a2 b2 of the endpoints of the transverse exis. One of the endpoints of the conjugate exis.

16.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And Find the eccentricity of the hyperbola with asymptotes 4x − 3y = 2. and 3x + 4y = 2 68  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And Find the equation of the hyperbola which has as its asymptotes and which passes through the origin. 4x + 3y + 1 = 0 and 3x − 4y + 7 = 0 69  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And Find the equation of the asymptotes of the hyperbola 3x2+ 10xy + 9y2+ 14x + 22y + 7 70 = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And If a hyperbola passing through the origin has as its asymptotes, then find the equation of its transvers and conjugate axes. and 3x − 4y − 1 = 0 71 4x − 3y − 6 = 0  Watch Free Video Solution on Doubtnut

17. CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And 72 Find the product of the length of perpendiculars drawn from any point on the hyperbola to its asymptotes. x2− 2y2− 2 = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And y2 x2 73 Find the angle between the asymptotes of the hyperbola . − = 1 16 9  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And y2 x2 Find the area of the triangle formed by any tangent to the hyperbola − = 1 74 a2 b2 with its asymptotes.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And 75 Find the asymptotes of the curve . xy − 3y − 2x = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And Show that the acute angle between the asymptotes of the hyperbola x2 y2 2cos−1( ), 1 is where is the eccentricity of the e = 1, (a2> b2), 76 − a2 b2 e hyperbola.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And A triangle has its vertices on a rectangular hyperbola. Prove that the orthocentre of the triangle also lies on the same hyperbola. 77  Watch Free Video Solution on Doubtnut

18. CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And If right angle at are three points on the hyperbola then prove that C, such that subtends a xy = c2 A,B, andC AB 78 is parallel to the normal to the hyperbola at point AB C.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And If asymptotes, then find the locus of the midpoint of is the perpendicular from a point on a rectangular hyperbola to its xy = c2 PN 79 PN  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And are two perpendicular chords of the rectangular hyperbola is the center of the rectangular hyperbola, then find the value of product of the slopes of and CP, CQ, CR, CS. and If xy = c2 PQ RS C . 80  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Miscellaneous A variable line . Prove that the locus of the centroid of triangle hyperbola passing through the origin. cuts the lines and OAB(O at points y = mx − 1 x = 2y y = − 2x AandB being the origin) is a 81  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 Two tangents to the hyperbola having cut the axes at four − = 1 82 m1andm2 a2 b2

19. concyclic points. Fid the value of m1m2.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: Let be a point on the hyperbola is nearest to the line P where is a parameter, such that a x2− y2= a2, P 83 Find the locus of y = 2x. P.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular Find the range of parameter for which a unique circle will pass through the points of intersection of the hyperbola equation of the circle. a x2− y2= a2 and the parabola Also, find the . y = x2 84  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And 2 2

20. y2 x2 Show that the midpoints of focal chords of a hyperbola lie on another 85 − = 1 a2 b2 similar hyperbola.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 y2 x2 x2 A tangent to the hyperbola cuts the ellipse = 1 at − + = 1 a2 b2 a2 b2 . Show that the locus of the midpoint of is 86 PandQ PQ 2 y2 y2 x2 x2 ( ) + = − . a2 b2 a2 b2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 Prove that the part of the tangent at any point of the hyperbola − = 1 a2 b2 87 intercepted between the point of contact and the transvers axis is a harmonic mean between the lengths of the perpendiculars drawn from the foci on the normal at the same point.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And If one of varying central conic (hyperbola) is fixed in magnitude and position, prove that the locus of the point of contact of a tangent drawn to it from a fixed point on the other axis is a parabole. 88  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 A transvers axis cuts the same branch of a hyperbola at − = 1 PandP' 89 a2 b2 PQ′= P′Q. and the asymptotes at and Q . Prove that and PQ = P'Q' Q'  Watch Free Video Solution on Doubtnut

21. CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: 1. If the distance between two parallel tangents drawn to the hyperbola 1 is 2, then their slope is equal 49 b. t d. none of these 90  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: If the distance between the foci and the distance between the two directricies of the x2 y2 91 hyperbola are in the ratio 3:2, then = 1 is (a) (b) (c) − b:a 1:√2 √3: √2 a2 b2 (d) 1:2 2:1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 π Atangent drawn to hyperbola at froms a triangle of area ) 3a2 P( − = 1 92 a2 b2 6 square units, with the coordinate axes, then the square of its eccentricity is  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular The length of the transverse axis of the rectangular hyperbola 18 (d) 9 is 6 (b) 12 (c) xy = 18 93  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular A straight line has its extremities on two fixed straight lines and cuts off from them a triangle of constant area Then the locus of the middle point of the line is c2 . 2 2 2 2xy = c2 94

22. (b) (d) none of these xy + c2= 0 4x2y2= c  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola e ′ If a variable line has its intercepts on the coordinate axes where eande′, 2 □ ande 2 95 are the eccentricities of a hyperbola and its conjugate hyperbola, then the line always touches the circle where x2+ y2= r2, r = 1 (b) 2 (c) 3 (d) cannot be decided  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: The (x − 3)2+ (y + 1)2= (4x + 3y)2 3x − 4y = 13 equation of the transvers is axis of the hyperbola (b) 96 x + 3y = 0 4x + 3y = 9 (d) 4x + 3y = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: The curve for which the length of the normal is equal to the length of the radius vector is/are (a) circles (b) rectangular hyperbola (c) ellipses (d) straight lines 97  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: The eccentricity of the conic represented by 1 is 1 (b) x2− y2− 4x + 4y + 16 = 0 98 (c) 2 (d) √2 2  Watch Free Video Solution on Doubtnut

23. CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: The equation 16x2− 3y2− 3y2− 32x + 12y − 44 = 0 99 represents a hyperbola. the length of whose transvers axis is the length of 4√3 √19 whose transvers axis is whose center is whose eccentricity is ( − 1,2) 4 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: If the vertex of a hyperbola bisects the distance between its center and the correspoinding focus, then the ratio of the square of its conjugate axis to the square of its transverse axis is 2 (b) 4 (c) 6 (d) 3 100  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Auxiliary Circle And Eccentric Angle The eccentricity of the hyperbola whose length of the latus rectum is equal to 8 and the length of its conjugate axis is equal to half of the distance between its foci, is : (1) 4 4 2 √3 101 (2) (3) (4) 3 √3 √3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 Let be the latus rectum through the focus of the hyperbola and − = 1 LL' a2 b2 be the farther vertex. If is equilateral, then the eccentricity of the A' A'LL'

24. 102 √3 + 1 √3 + 1 ( (d) ) hyperbola is (axes are coordinate axes). (b) √3 √2 (√3 + 1) √3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: Show that the equation 9 hyperbola. Find the coordinates of the centre, lengths of the axes, eccentricity, latus- rectum, coordinates of foci and vertices, equations of the directrices of the hyperbola. represents a x2− 16y2− 18x + 32y − 151 = 0 103  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Conjugate The eccentricity of the conjugate hyperbola of the hyperbola 4 is 2 (b) x2− 3y2= 1 104 (c) 4 (d) 2√3 5  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: The equation of the transvers and conjugate axes of a hyperbola are, respectively, and , and their respective lengths are x + 2y − 3 = 0 2x − y + 4 = 0 2 and √2 3 (x + 2y − 3)2− (2x − y + 4)2= 1 The equation of the hyperbola is 2√3. 105 5 5 2 3 (x − y − 4)2− (x + 2y − 3)2= 1 5 2(2x − y + 4)2− 3(x + 2y − 3)2= 1 2(x + 2y − 3)2− 3(2x − y + 4)2= 1 5  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: The locus of the point of intersection of the lines √3x − y − 4√3t = 0&√3tx + ty 106 − 4√3 = 0 (where t is a parameter) is a hyperbola whose eccentricity is:  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: 2 2 α

25. lf the eccentricity of the hyperbola the ellipse x2(sec)2α + y2= 25 is times the eccentricity of √3 x2− y2(sec)α= 5 107 , then a value of is: α  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Auxiliary Circle And Eccentric Angle y2 x2 With one focus of the hyperbola as the centre, a circle is drawn which − = 1 108 9 16 is tangent to the hyperbola with no part of the circle being outside the hyperbola. The radius of the circle is  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 If is tangent to the hyperbola , then is equal to a2− b2 ax + by = 1 − = 1 a2 b2 109 1 (a) (b) (c) (d) none of these a2e2 b2e2 a2e2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular The locus of a point whose chord of contact with respect to the circle a tangent to the hyperbola parabola is x2+ y2= 4 110 is a/an (a)ellipse (b) circle (c)hyperbola (d) xy = 1

26.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: The sides x2 of a touch the conjugate hyperbola of the hyperbola x2 ACandAB ABC y2 y2 111 . If the vertex = 1 lies on the ellipse A , then the side − + = 1 a2 BC b2 a2 b2 must touch parabola (b) circle hyperbola (d) ellipse  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 The tangent at a point on the hyperbola passes through the point − = 1 P a2 b2 112 and the normal at passes through the point . Then the (0, − b) (2a√2,0) P eccentricity of the hyperbola is 2 (b) (c) 3 (d) √2 √3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: If values of a, for which the line 16x2− 9y2= 144 touches the hyperbola x2− (a1+ b1)x − 4 = 0 y = ax + 2√5 113 are the roots of the equation is a1+ b1 , then the values of  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola y2 x2 If the angle between the asymptotes of hyperbola is and the 1200 − = 1 a2 b2

27. 114 product of perpendiculars drawn from the foci upon its any tangent is 9, then the locus of the point of intersection of perpendicular tangents of the hyperbola can be (a) (b) (c) x2+ y2= 6 x2+ y2= 9 x2+ y2= 3 (d) x2+ y2= 18  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And y2 x2 The coordinates of a point on the hyperbola which s nearest to the = 1 − 115 24 18 line are (6, 3) (b) (d) 3x + 2y + 1 = 0 ( − 6, − 3) 6, − 3) ( − 6,3)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Position Of A Point (H K) With Respect To A Hyperbola The number of possible tangents which can be drawn to the curve which are perpendicular to the straight line 4 4x2− 9y2= 36, 116 , is zero (b) 1 (c) 2 (d) 5x + 2y − 10 = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: Locus of the feet of the perpendiculars drawn from either foci on a variable tangent to the hyperbola is 16y2− 9x2= 1 117  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: The locus of the foot of the perpendicular from the center of the hyperbola on xy = 1 7 (x2− y2) =1 a variable tangent is (b) (d) (x2− y2) = 4xy 9(x2− y2) = 118 144 1 (x2− y2) = 16  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: The locus of the foot of the perpendicular from the center of the hyperbola on xy = 1 7 (x2− y2) =1 a variable tangent is (b) (d) (x2− y2) = 4xy 9(x2− y2) = 119 144 1 (x2− y2) = 16  Watch Free Video Solution on Doubtnut

28. CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: Nis the foot of the perpendicular from P on the transverse os Pisapoint on the hyperbola ais The tangent tothe laat P meets the transverse axis at T.Ifois the centre of the hy the OLON is equal to: (D)bela LA) 120  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 The tangent at a point on the hyperbola meets one of the directrix − = 1 P a2 b2 π π 121 at If subtends an angle at the corresponding focus, then θ (b) (c) θ = F. PF 4 2 3π (d) π 4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular The locus of a point, from where the tangents to the rectangular hyperbola contain an angle of , is x2− y2= a2 450 2+ a2(x2− y2) = 4a2 (x2+ y2) 122 2+ 4a2(x2− y2) = 4a2 2+ 4a2(x2− y2) = 4a2 2(x2+ y2) (x2+ y2) + a2(x2− y2) = a4 (x2+ y2)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: If tangents x2 are drawn from a variable point PQandPR to thehyperbola P y2 so that the fourth vertex of parallelogram lies − = 1, (a > b), S PQSR 123 a2 b2 on the circumcircle of triangle , then the locus of PQR is (b) x2+ y2= b2 P 2 2 2 2 2 2 2

29. (d) none of these x2+ y2= a2x2+ y2= a2− b2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 The number of points on the hyperbola from which mutually = 3 − a2 b2 124 perpendicular tangents can be drawn to the circle (d) 4 is/are 0 (b) 2 (c) 3 x2+ y2= a2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 A normal to the hyperbola has equal intercepts on the positive x- and − = 1 4 1 125 y2 x2 y-axis. If this normal touches the ellipse , then is equal to 5 a2+ b2 + = 1 a2 b2 (b) 25 (c) 16 (d) none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular c If the normal to the given hyperbola at the point meets the curve again at ) (ct, 126 t c then (a) (b) (c) (d) t3t′= 1 t3t′= − 1 (ct′, tt′= 1 tt′= − 1 ), t′  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular If the sum of the slopes of the normal from a point to the hyperbola is xy = c2 P + 2 2 2 2

30. 127 equal to xy = λc2 , then the locus of point is (a) (b) (c) λ(λ ∈ R+) x2= λc2 y2= λc2 P (d) none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And If a ray of light incident along the line x2 gets reflected from 3x + (5 − 4√2)y = 15 y2 128 the hyperbola , then its reflected ray goes along the line. = 1 − 16 9 (b) (d) none of these x√2 − y + 5 = 0 √2y − x + 5 = 0√2y − x − 5 = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola y2 x2 Let any double ordinate of the hyperbola be produced on = 1 − PNP' 25 16 . P 129 ′Q both sides to meet the asymptotes in (c) 41 (d) none of these . Then is equal to 25 (b) 16 QandQ' PQ  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola For hyperbola whose center is at (1, 2) and the asymptotes are parallel to lines and , the equation of the hyperbola passing through (2, 4) is none of these (2x + 3y − 8)(x + 2y − 5) = 30 2x + 3y = 0 x + 2y = 1 130 (2x + 3y − 5)(x + 2y − 8) = 40 (2x + 3y − 8)(x + 2y − 8) = 40  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And The chord of contact of a point angle. Then the point asymptotes (d) none of these w.r.t a hyperbola and its auxiliary circle are at right lies on conjugate hyperbola one of the directrix one of the P 131 P  Watch Free Video Solution on Doubtnut

31. CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola If the intercepts made by tangent, normal to a rectangular are and with y-axis are a1,a2 with x-axis x2− y2= a2 132 then b1,b2 a1, a2+ b1b2=  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola If then the value of for which (d) 18 −22 is the equation of the hyperbola S = 0 , x2+ 4xy + 3y2− 4x + 2y + 1 = 0 133 represents its asymptotes is (b) (c) S + K = 0 20 −16 k  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: If two distinct tangents can be drawn from the Point x2 y2 on different branches of (α,2) 134 |α| <3 |α| >2 the hyperbola then 1) 2) 3) 4) − = 1 |α| > 3 α = 1 9 16 2 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola A hyperbola passes through (2,3) and has asymptotes . Then, 77x − 21y − 265 = 0 21x + 77y − 265 = 0 and 3x − 4y + 5 = 0 the equation of its transverse 21x − 77y − 265 = 0 axis is 12x + 5y − 40 = 0 135 21x − 77y + 265 = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: (1, 2)

32. From a point is drawn to each arm of the hyperbola. If the equations of the asymptotes of hyperbola are and H √3x − y + 5 = 0 √3x + y − 1 = 0 2 √2 √3 , two tangents are drawn to a hyperbola in which one tangent P(1, 2) H 136 , then the eccentricity of is (a) 2 H (b) (c) (d) √3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola The 2x2+ 5xy + 2y2+ 4x + 5y = 0 2x2+ 5xy + 2y2+ 4x + 5y − 2 = 0 2x2+ 5xy + 2y2= 0 combined equation of the is asymptotes 2x2+ 5xy + 2y2+ 4x + 5y + 2 = 0 of the hyperbola 137 none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola The asymptotes of the hyperbola and x + h = 0 y + k = 0 x − k = 0 are and and xy = hx + ky x − k = 0 y − h = 0 y − h = 0 138 and y + h = 0 x + k = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 The asymptote of the hyperbola form with ans tangen to the + = 1 139 a2 b2 hyperbola triangle whose area is in magnitude then its eccentricity is: a2tanλ  Watch Free Video Solution on Doubtnut

33. CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola The center of a rectangular hyperbola lies on the line asymptotes is 3x + 6y − 5c = 0 3x − 6y − c = 0 If one of the y = 2x. 6x + 3y − 4c = 0 140 , then the other asymptote is (d) none of these (b) x + y + c = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular The equation of a rectangular hyperbola whose asymptotes are and passing through (7,8) is xy − 3y + 5x − 3 = 0xy − 3y + 5x + 3 = 0 and x = 3 y = 5 141 xy − 3y + 5x + 3 = 0 xy + 3y + 4x + 3 = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: If the foci of a hyperbola lie on equation of the hyperbola, given that it passes through (3, 4), is (a) 5 2x2− 2y2+ 5xy + 5 = 0 and one of the asymptotes is then the y = x y = 2x, 142 (b) (c) x2− y2− xy + 5 = 0 2 (d)none of these 2x2+ 2y2+ 5xy + 10 = 0  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular (x − 1)2+ (y + 2)2= r2 (x-1)(y-2)=5 and centroid of intersect at four points A, B, C, D and if , then locus of D is y = 3x − 4 143 lies on line △ ABC  Watch Free Video Solution on Doubtnut

34. CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular If tangents lying on the rectangular hyperbola and are dawn to variable circles having radius and the center , then the locus of the circumcenter of xy = 1 xy =1 OQ OR r 144 triangle is being the origin). (O (b) (d) none of these xy = 4 4xy = 1 OQR  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular The equation of conjugate axis of the hyperbola (b) y + x = 3 y + x = 7 y − x = 3 is xy − 3y − 4x + 7 = 0 145 (d) none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola If that of the conjugate axis is 6, and then the area of quadrilateral are the foci of the hyperbola whose length of the transverse axis is 4 and are the foci of the conjugate hyperbola, is 24 (b) 26 (c) 22 (d) none of these S1S3S2S4 S1andS2 146 S3andS4  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular Suppose the circle having equation at xy = 1 x2+ y2− 3 + λ(xy − 1) = 0,λ ∈ R, intersects the rectangular hyperbola The represents. a pair of lines through the for a circle for any λ = − 3 λ ∈ R x2+ y2= 3 A,B, C, andD. points equation 147 origin for through an ellipse through for a parabola λ = − 3 A,B,C,andD A, B, C, andD λ = − 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And y2 x2 If two points on the hyperbola , P&Q whose centre is C be such that − = 1 a2 CQ and a < b b2 148 CP is perpendicularal 1 to 1 ,then prove that 1 1 1 . + = − CP2 CQ2 a2 b2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular The equation to the chord joining two points on the rectangular (x1, y1)and(x2,y2) y y x x

35. y y x x 149 hyperbola is: xy = c2 + = 1 + = 1 y1+ y2 y1− y2 x1+ x2 x1− x2 y y x x (d) + = 1 + = 1 y1+ y2 y1− y2 x1+ x2 x1− x2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular If P(x1,y1), Q(x2,y2), 150 R(x3, y3) and S(x4,y4) are four concyclic points on the rectangular hyperbola ) and coordinates of the orthocentre ofthe triangle , then xy = c2 is PQR  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular The chord the midpoint of right-angled (d) right isosceles of the rectangular hyperbola and is the origin. Then PQ; O meets the axis of at is equilateral (b) isosceles ΔACO is xy = a2 A;C PQ x 151  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular The curve the distance between the point of contacts is 1 (b) 2 (c) and the circle touch at two points. Then (d) none of these 2√2 x2+ y2= 1 xy = c, (c > 0), 152  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular = 2 +

36. Let and at four points: OP2+ OP2+ OR2+ OS2 be a curve which is the locus of the point of intersection of lines A circle . If is x = 2 + m C 153 s ≡ (x − 2)2+ (y + 1)2= 25 intersects the curve my = 4 − m. C is center of the curve then P,Q,R,andS C, O  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And The ellipse angles. Then the equation of the circle through the points of intersection of two conics is and the hyperbola intersect at right 4x2+ 9y2= 36 a2x2− y2= 4 154  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And The locus of the point which is such that the chord of contact of tangents drawn from it x2 y2 155 to the ellipse forms a triangle of constant area with the coordinate + = 1 a2 b2 axes is a straight line (b) a hyperbola an ellipse (d) a circle  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular The angle between the lines joining origin to the points of intersection of the line 2 √3 π 6tan−1( ) and the curve is (b) ) y2− x2= 4 tan−1( √3x + y = 2 156 2 √3 π (d) 2  Watch Free Video Solution on Doubtnut

37. CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And y2 x2 A variable chord of the hyperbola subtends a right angle at − = 1, (b > a), 157 a2 b2 the center of the hyperbola if this chord touches. a fixed circle concentric with the hyperbola a fixed ellipse concentric with the hyperbola a fixed hyperbola concentric with the hyperbola a fixed parabola having vertex at (0, 0).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: The equation ∣∣∣√x2+ (y − 1)2− √x2+ (y + 1)2∣∣∣ 158 will represent a hyperbola for = K (b) (d) K ∈ (0,2) K ∈ ( − 2,1) K ∈ (1,∞) K ∈ (0,∞)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: If 3x4− 2(19y + 8)x2 then the equation x, y ∈ R, + (361y2+ 2(100 + y4) + 64, 159 = 2(190y + 2y2) represents in rectangular Cartesian system a/an (a)parabola (b) hyperbola (c)circle (d) ellipse  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: If are the foci of a hyperbola passing through the origin, then (5, 12)and(24,7) 160 e =√386 e =√386 LR =121 LR =121 (b) (d) 12 13 6 3  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: If (5, 12) and (24, 7) are the foci of an ellipse passing through the origin, then find the eccentricity of the ellipse. 161  Watch Free Video Solution on Doubtnut

38. CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And For which of the hyperbolas, can we have more than one pair of perpendicular x2 y2 − = − 1 x2 4 9 y2 162 tangents? (b) (d) x2− y2= 4 − = 1 xy = 44 4 9  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: Show that the equation 9 hyperbola. Find the coordinates of the centre, lengths of the axes, eccentricity, latus- rectum, coordinates of foci and vertices, equations of the directrices of the hyperbola. represents a x2− 16y2− 18x + 32y − 151 = 0 163  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: y2 y2 x2 x2 If the foci of coincide with the foci of = 1 and the − + = 1 a2 b2 25 9 164 eccentricity of the hyperbola is 2, then hyperbola the center of the director circle is (0, 0). the length of latus rectum of the hyperbola is 12 there is no director circle to the a2+ b2= 16  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular If the circle P(x1,y1), Q(x2,y2), R(x3,y3), y1+ y2+ y3+ y4= 0 x1x2x3x4= C4y1y2y3y4= C4 intersects the hyperbola and at four points x2+ y2= a2 xy = c2 165 then S(x4, y4), x1+ x2+ x3+ x4= 0  Watch Free Video Solution on Doubtnut

39. CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: 3y dx The differential equation represents a family of hyperbolas (except when = 2x dy 166 √3 √5 3√2 √5 it represents a pair of lines) with eccentricity. (b) (d) 5 5 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular Circle are drawn on the chords of the rectangular hyperbola line as diameters. All such circles pass through two fixed points whose coordinates are (b) (c) (2,2) (2, − 2) ( − 2, 2) parallel to the xy = 4 167 y = x (d) ( − 2, − 2)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: The equation (x − α)2+ (y − β)2 = k(lx + my + n)2 168 − 1 represents a parabola for an ellipse for `0(1^2+m^2)^(-1) k < (l2+ m2) k=0` ap∮ ∘ ≤ f or  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular The lines parallel to the normal to the curve 3x − 4y + 5 = 0 4x + 3y + 5 = 0 is/are (b) xy = 1 3x + 4y + 5 = 0 169 (d) 3y − 4x + 5 = 0

40.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And y2 x2 From the point (2, 2) tangent are drawn to the hyperbola Then the − = 1. 170 16 9 point of contact lies in the first quadrant (b) second quadrant third quadrant (d) fourth quadrant  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: If the two intersecting lines intersect the hyperbola and neither of them is a tangent to it, then the number of intersecting points are 1 (b) 2 (c) 3 (d) 4 171  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 For the hyperbola , let be the number of points on the plane n − = 1 a2 b2 172 through which perpendicular tangents are drawn. If e n0sqrt(2)` None of these If `n >1,t h n = 1, the ≠ = √2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular If the normal at and gandC PG − Pg to the rectangular hyperbola is the center of the hyperbola, then (d) Gg = 2PC meets the axes at (b) PG = PC x2− y2= 4 P G 173 Pg = PC  Watch Free Video Solution on Doubtnut

41. CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: Statement 1 : The asymptotes of hyperbolas bisectors of the transvers and conjugate axes of the hyperbolas. Statement 2 : The transverse and conjugate axes of the hyperbolas are the bisectors of the asymptotes. and are the 3x + 4y = 2 4x − 3y = 5 174  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 Statement 1 : Every line which cuts the hyperbola at two distinct = 1 − 175 4 16 points has slope lying in hyperbola lies in Statement 2 : The slope of the tangents of a ( − 2,2). ( − ∞, − 2) ∪ (2,∞).  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: Statement 1 : A bullet is fired and it hits a target. An observer in the same plane heard two sounds: the crack of the rifle and the thud of the bullet striking the target at the same instant. Then the locus of the observer is a hyperbola where the velocity of sound is smaller than the velocity of the bullet. Statement 2 : If the difference of distances of a point from two fixed points is constant and less than the distance between the fixed points, then the locus of is a hyperbola. P 176 P  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 If hyperbola passes through the foci of the ellipse a . Its transverse + = 1 25 16 and conjugate axes coincide respectively with the major and minor axes of the ellipse and if the product of eccentricities of hyperbola and ellipse is 1 then the equation of x2 y2 177 y2 x2 hyperbola is b. the equation of hyperbola is = 1 c. focus − − = 1 9 16 9 25 of hyperbola is (5, 0) d. focus of hyperbola is (5√3, 0)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Position Of A Point (H K) With Respect To A Hyperbola y2 x2 If a point lies in the shaded region (x1, y1) , shown in the figure, then − = 1 a2 b2 y2 y2 x2 x2 178 Statement 2 : If < 0 lies outside the hyperbola P(x1,y1) − − = 1 a2 b2 a2 b2 x12 y12 , then − < 1 a2 b2

42.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: Statement 1 : The equations of tangents to the hyperbola parallel to the line a given slope, two parallel tangents can be drawn to the hyperbola. which is 2x2− 3y2= 6 y = 3x + 5. 179 are and Statement 2 : For y = 3x + 4 y = 3x − 5  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: Number of points from where perpendicular tangents can be drawn to the curve x2 y2 180 is − = 1 16 25  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 Statement 1 : If from any point on the hyperbola , P(x1, y1) − = − 1 a2 b2 y2 x2 tangents are drawn to the hyperbola then the corresponding chord − = 1, 181 a2 b2 y2 x2 of contact lies on an other branch of the hyperbola Statement 2 : − = − 1 a2 b2 From any point outside the hyperbola, two tangents can be drawn to the hyperbola.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular Statement 1 : If (3, 4) is a point on a hyperbola having foci (3, 0) and length of the transverse axis being 1 unit, then can take the value 0 or 3. Statement , the (λ, 0) λ 182 ′ ∣ ∣

43. 2 : transverse axis, and is any point on the hyperbola. P where are the two foci, is the length of the ∣∣S′P − SP∣∣ = 2a, 2a SandS'  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Miscellaneous are two fixed points and P 1s a point such that Let S be the circle , then the number of points P satisfying is x2+ y2= r2 A( − 2,0) and B(2,0) , then match the following. If PA − PB = 2 x2+ y2= r2 PA − PB = 2 r = 2 183 and lying on  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: The eccentricity of the hyperbola ∣∣∣√(x − 3)2+ (y − 2)2 184 −√(x + 1)2+ (y + 1)2∣∣∣= 1 is ______  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 If is tangent to the hyperbola having eccentricity 5, y = mx + c − = 1, 185 a2 b2 then the least positive integral value of is_____ m  Watch Free Video Solution on Doubtnut

44. CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular Consider the graphs of constant and , where A is a positive y = Ax2and y2+ 3 = x2+ 4y 186 .Number of points in which the two graphs intersect, is x,y ∈ R  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: nd are inclined at avgicsTangents are drawn from the point and are inclined atv angle , prove that . β2= 2α2− 7 to the hyperbola (α,β) 187 to the x-axis.If 3x2− 2y2= 6 θ and ϕ tanθ.tanϕ = 2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 If tangents drawn from the point to the hyperbola (a,2) are − = 1 188 16 9 perpendicular, then the value of is _____ a2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular If the hyperbola center keeping the axis intact, then the equation of the hyperbola is is equal to___________ a2 is rotated by in the anticlockwise direction about its x2− y2= 4 450 189 where xy = a2,  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And The area of triangle formed by the tangents from the point (3, 2) to the hyperbola and the chord of contact w.r.t. the point (3, 2) is_____________ x2− 9y2= 9 190  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola e ′ If a variable line has its intercepts on the coordinate axes where eande′, 2 □ ande 2 191 are the eccentricities of a hyperbola and its conjugate hyperbola, then the line always touches the circle where x2+ y2= r2, r = 1 (b) 2 (c) 3 (d) cannot be decided

45.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE GEOMETRY_HYPERBOLA_Comparison Of Hyperbola And Its Conjugate Hyperbola If the vertex of a hyperbola bisects the distance between its center and the correspoinding focus, then the ratio of the square of its conjugate axis to the square of its transverse axis is 2 (b) 4 (c) 6 (d) 3 192  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: If the distance between two parallel tangents having slope x2 y2 drawn to the hyperbola m 193 is 2, then the value of is_____ − = 1 2|m| 9 49  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular If is the length of the latus rectum of the hyperbola for which the equations of asymptotes and which passes through the point (4, 6), then the value L are x = 3andy = 2 L 194 of is_____ √2  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And y2 x2 If the chord of the hyperbola subtends a right xcosα + ysinα = p − = 1 16 18 195 angle at the center, and the diameter of the circle, concentric with the hyperbola, to d which the given chord is a tangent is then the value of is__________ d, 4

46.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 π Atangent drawn to hyperbola at froms a triangle of area ) 3a2 P( − = 1 196 a2 b2 6 square units, with the coordinate axes, then the square of its eccentricity is  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: lf the eccentricity of the hyperbola the ellipse x2(sec)2α + y2= 25 is times the eccentricity of √3 x2− y2(sec)α= 5 197 , then a value of is: α  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And The locus of a point whose chord of contact with respect to the circle a tangent to the hyperbola is a/an ellipse (b) circle hyperbola (d) parabola xy = 1 is x2+ y2= 4 198  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And 2 2

47. 199 y2 x2 Tangents are drawn from any point on the hyperbola to the circle = 1 − 9 4 . Find the locus of the midpoint of the chord of contact. x2+ y2= 9  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular 1 1 An ellipse has eccentricity and one focus at the point . Its one directrix is P( ,1) 2 2 200 the comionand tangent nearer to the point the P to the hyperbolaof the circle .Find the equation of the ellipse. x2+ y2= 1 and x2− y2= 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 The equation represents (a)an ellipse (b) a hyperbola 201 − = 1, r > 1, 1 − r 1 + r (c)a circle (d) none of these  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: Each of the four inequalities given below defines a region in the xy plane. One of these four regions does nothave the following property. For any two points y1+ y2 x1+ x2 202 in the region the piont is also in the ( ) (x1,y2) and (y1,y2) ⋅ 2 2 region. The inequality defining this region is  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: 203 The equation, represents 2x2+ 3y2− 8x − 18y + 35 = K  Watch Free Video Solution on Doubtnut

48. CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: θ + ϕ =π Let and ( where be two points P(asecθ,b tanθ) Q(a secϕ, btanϕ) 2 y2 x2 204 on the hyperbola . If (h, k) is the point of intersection of the normals at − = 1 a2 b2 P and Q then k is equal to  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Chords Of Contact GEOMETRY_HYPERBOLA_Chords And If the corresponding pair of tangents is is the chord of contact of the hyperbola then the equation of x2− y2= 9 x = 9 205  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: 206 Which of the following is independent of in the hyperbola `(0 < alpha  Watch Free Video Solution on Doubtnut α CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: If the line touches the hyperbola , then the point of x2− 2y2= 4 2x + √6y = 2 207 1 1 contact is (b) (d) ( − 5,2√6) ( ) ( − 2, √6) , (4, − √6) 2 √6  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: A hyperbola having the transverse axis of length is confocal with the ellipse 2sinθ 2 2 2 2 2 2

49. 208 . Then its equation is 3x2+ 4y2= 12 x2sec2θ − y2cosec2θ = 1 x2sin2θ − y2cos2θ = 1 x2cos2θ − y2sin2θ = 1 x2cosec2θ − y2sec2θ = 1  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: Consider a branch of the hypetrbolar vertex at the point A. Let B be one of the end points of its latus rectum. If C is the focus of the hyperbola nearest to the point A, then the area of the triangle ABC is with x2− 2y2− 2√2x − 4√2y − 6 = 0 209  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: Let (ax2+ by2+ c)(x2− 5xy + 6y2) = 0 be nonzero real numbers. represents. four straight lines, when Then the equation aandb are of the same sign. two straight lines and a circle, when is of sign opposite to that two straight lines and a hyperbola, when the same sign and is of sign opposite to that of a circle and an ellipse, when are of the same sign and is of sign opposite to that of aandb c and and are of 210 c = 0 a,b a = b c a aandb c a a  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 Let P(6,3) be a point on the hyperbola parabola If the normal at the − = 1 211 a2 b2

50. a b point intersects the x-axis at (9,0), then the eccentricity of the hyperbola is  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Hyperbola GEOMETRY_HYPERBOLA_Rectangular y2 x2 If hyperbola passes through the foci of the ellipse a . Its transverse + = 1 25 16 and conjugate axes coincide respectively with the major and minor axes of the ellipse and if the product of eccentricities of hyperbola and ellipse is 1 then the equation of x2 y2 212 y2 x2 hyperbola is b. the equation of hyperbola is = 1 c. focus − − = 1 9 16 9 25 of hyperbola is (5, 0) d. focus of hyperbola is (5√3, 0)  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 1 GEOMETRY_HYPERBOLA_Hyperbola: 7. An ellipse intersects the hyperbola 2x2-2y 1 orthogonally. The eccentricity of the ellipse is reciprocal to that of the (2009) hyperbola. If the axes of the ellipse are along the coordinate axes, then (b) the foci of ellipse are (+1, 0) (a) equation of ellipse is x2+ 2y2 2 (d) the foci of ellipse are (t 2, 0) (c) equation of ellipse is x2 2y 213  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 let the eccentricity of the hyperbola be reciprocal to that of the ellipse − = 1 214 a2 b2 if the hyperbola passes through a focus of the ellipse then: x2+ 4y2= 4.  Watch Free Video Solution on Doubtnut CENGAGE_MATHS_COORDINATE Definition 2 GEOMETRY_HYPERBOLA_Hyperbola: y2 x2 Tangents are drawn to the hyperbola parallet to the sraight line = 1 − 215 . The points of contact of the tangents on the hyperbola are 9 2x − y = 1 4  Watch Free Video Solution on Doubtnut