Hyperbola PYQs Set 3 (PDF)

Set 3 of 11 · Questions 61-90 of 302

In the print window choose “Save as PDF”. The file prints without a web address or date line.

Mere Teacher · Previous Year Questions JEE Main Mathematics · Hyperbola · Set 3 (Q61-90)
Scan or visit mereteacher.com for free PYQ practice, timed CBT tests, step-by-step solutions and BroBot AI. Answers and solutions are free with a sign-up.
61. $If a hyperbola passes through the point P (10, 16) and it has vertices at (\pm 6, 0) then the equation of the normal to it at P is:$ (1) $2 x + 5 y = 100$ (2) $2 x + 5 y = 10$ (3) $2 x - 5 y = 100$ (4) $5 x + 2 y = 100$ 62. $Let A (θ_{1}) and B (θ_{2}) be two points on the hyperbola \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1 and S be the focus of the hyperbola. If A, S, B are collinear and a cos (\frac{θ _{1} + θ _{2}}{2}) = k cos (\frac{θ _{1} - θ _{2}}{2}) then k =$ (1) $a^{2} + b^{2}$ (2) $a^{2} + b^{2}$ (3) $a^{2} - b^{2}$ (4) $a + b$ 63. $If the lines kx + 2 y - 4 = 0 and 5 x - 2 y - 4 = 0 are conjugate with respect to the circle x^{2} + y^{2} - 2 x - 2 y + 1 = 0, then k is equal to$ (1) $0$ (2) $1$ (3) $2$ (4) $3$ 64. $If the circle x^{2} + y^{2} = a^{2} intersects the hyperbola xy = c^{2} in four points P (x_{1}, y_{1}), Q (x_{2}, y_{2}), R (x_{3}, y_{3}) and S (x_{4}, y_{4}), then$ (1) $y_{1} y_{2} y_{3} y_{4} = 2 c^{4}$ (2) $x_{1} + x_{2} + x_{3} + x_{4} = 0$ (3) $y_{1} + y_{2} + y_{3} + y_{4} = 2$ (4) $x_{1} x_{2} x_{3} x_{4} = 2 c^{4}$ 65. $If (4, 7) and (- 2, - 1) are ends of a diameter of a circle which intersects X -axis at A and B, then AB is equal to$ (1) $5$ (2) $6$ (3) $7$ (4) $8$ 66. $If 5 x + 6 y - 34 = 0 and 2 x + y + c = 0 are conjugate lines with respect to the circle x^{2} + y^{2} - 8 x - 10 y + 25 = 0 then the point on the line 2 x + y + c = 0 is$ (1) $(3, 3)$ (2) $(2, 4)$ (3) $(1, - 5)$ (4) $(- 2, - 2)$ 67. $The points of intersection of the parabolas y^{2} = 5 x and x^{2} = 5 y lie on the line$ (1) $x + y = 10$ (2) $x - 2 y = 0$ (3) $x - y = 0$ (4) $2 x - y = 0$ 68. $The circle touching the coordinate axes with its centre lying on x - 2 y - 3 = 0 is$ (1) $x^{2} + y^{2} - 2 x + 2 y + 1 = 0$ (2) $x^{2} + y^{2} + 2 x - 2 y + 1 = 0$ (3) $x^{2} + y^{2} + 6 x + 6 y - 9 = 0$ (4) $x^{2} + y^{2} - 6 x - 6 y + 9 = 0$ 69. $Let origin be the centre, (\pm 3, 0) be the foci and \frac{3}{2} be the eccentricity of a hyperbola. Then the line 2 x - y - 1 = 0$ (1) $intersects the hyperbola at two points$ (2) $does not intersect the hyperbola$ (3) $touches the hyperbola$ (4) $passes through the vertex of the hyperbola$ 70. $Find the equation of the circle which passes through the point (1, 2) and the points of intersection of the circles x^{2} + y^{2} - 8 x - 6 y + 21 = 0 and x^{2} + y^{2} - 2 x - 15 = 0$ (1) $x^{2} + y^{2} - 18 x - 12 y + 27 = 0$ (2) $2 x^{2} + y^{2} - 18 x - 12 y + 27 = 0$ (3) $3 x^{2} + y^{2} - 18 x - 12 y + 27 = 0$ (4) $4 x^{2} + y^{2} - 18 x - 12 y + 27 = 0$ 71. $The length of the chord of the circle x^{2} + y^{2} + 3 x + 2 y - 8 = 0 intercepted by the y axis is$ (1) $3$ (2) $8$ (3) $9$ (4) $6$ 72. $Consider the circle x^{2} + y^{2} - 6 x + 4 y = 12 . The equations of a tangent of this circle that is parallel to the line 4 x + 3 y + 5 = 0 is$ (1) $4 x + 3 y + 10 = 0$ (2) $4 x + 3 y - 9 = 0$ (3) $4 x + 3 y + 9 = 0$ (4) $4 x + 3 y - 31 = 0$ 73. $If \frac{x ^{2}}{36} - \frac{y ^{2}}{k ^{2}} = 1 is a hyperbola, then which of the following statements can be true?$ (1) $(- 3, 1) lies on the hyperbola$ (2) $(3, 1) lies on the hyperbola$ (3) $(10, 4) lies on the hyperbola$ (4) $(5, 2) lies on the hyperbola$ 74. $Equation of circle with centre (- a, - b) and radius a^{2} - b^{2} is$ (1) $x^{2} + y^{2} + 2 ax + 2 by + 2 b^{2} = 0$ (2) $x^{2} + y^{2} - 2 ax - 2 by - 2 b^{2} = 0$ (3) $x^{2} + y^{2} - 2 ax - 2 by + 2 b^{2} = 0$ (4) $x^{2} + y^{2} - 2 ax + 2 by + 2 a^{2} = 0$ 75. $The equation of the circle passing through (1, 1) and through the points of intersection of the circles x^{2} + y^{2} + 13 x - 3 y = 0 and 2 x^{2} + 2 y^{2} + 4 x - 7 y - 25 = 0 is$ (1) $4 x^{2} + 4 y^{2} + 30 x - 13 y - 25 = 0$ (2) $2 x^{2} + 2 y^{2} + 15 x - 19 y = 0$ (3) $4 x^{2} + 4 y^{2} + 25 x + 12 y - 45 = 0$ (4) $4 x^{2} + 4 y^{2} + 13 x - 30 y + 9 = 0$ 76. $The equation of a tangent to the hyperbola 4 x^{2} - 5 y^{2} = 20 parallel to the line x - y = 2 is:$ (1) $x - y + 1 = 0$ (2) $x - y + 7 = 0$ (3) $x - y + 9 = 0$ (4) $x - y - 3 = 0$ 77. $Let x - 4 = 0 be the radical axis of two circles which are intersecting orthogonally. If x^{2} + y^{2} = 36 is one of those circles, then the other circle is$ (1) $x^{2} + y^{2} - 16 x + 36 = 0$ (2) $x^{2} + y^{2} - 18 x + 36 = 0$ (3) $x^{2} + y^{2} - 18 x + 24 = 0$ (4) $x^{2} + y^{2} - 6 x + 8 y + 36 = 0$ 78. $If the line 2 x + 3 y + n = 0 is a tangent to the parabola y^{2} = 8 x, then the equation of the normal drawn at the point (2 n, 4 n) to the parabola y^{2} = 8 x is$ (1) $x - 3 y + 18 = 0$ (2) $3 x + 2 y - 30 = 0$ (3) $3 x + y - 66 = 0$ (4) $2 x - 3 y + 6 = 0$ 79. $If 2 x^{2} + 2 y^{2} + 4 x + 5 y + 1 = 0 and 3 x^{2} + 3 y^{2} + 6 x - 7 y + 3 k = 0 are orthogonal, then value of k is$ (1) $\frac{17}{12}$ (2) $\frac{12}{17}$ (3) $\frac{- 12}{17}$ (4) $\frac{- 17}{12}$ 80. $The combined equation of the lines passing through the origin and having slopes \frac{2}{3} and \frac{- 2}{3} is$ (1) $2 x^{2} - 9 y^{2} = 0$ (2) $4 x^{2} - xy - 9 y^{2} = 0$ (3) $4 x^{2} - 9 y^{2} = 0$ (4) $4 x^{2} + xy - 9 y^{2} = 0$ 81. $Let P (h, k) be a point on the curve y = x^{2} + 7 x + 2, nearest to the line, y = 3 x - 3 . Then the equation of the normal to the curve at P is:$ (1) $x - 3 y - 11 = 0$ (2) $x - 3 y + 22 = 0$ (3) $x + 3 y - 62 = 0$ (4) $x + 3 y + 26 = 0$ 82. $x^{2} + y^{2} - 6 x - 6 y + 4 = 0, x^{2} + y^{2} - 2 x - 4 y + 3 = 0, x^{2} + y^{2} + 2 kx + 2 y + 1 = 0 . If the radical centre of the above three circles exists, then which of the following cannot be the value of k ?$ (1) $2$ (2) $5$ (3) $1$ (4) $4$ 83. $Let 0 < θ < \frac{π}{2} . If the eccentricity of the hyperbola \frac{x ^{2}}{cos ^{2} θ} - \frac{y ^{2}}{sin ^{2} θ} = 1 is greater than 2, then the length of its latus rectum lies in the interval:$ (1) $(3, \infty)$ (2) $(\frac{3}{2}, 2)$ (3) $(2, 3)$ (4) $(1, \frac{3}{2})$ 84. $The equation of a tangent to the hyperbola 16 x^{2} - 25 y^{2} - 96 x + 100 y - 356 = 0 which makes an angle 45^{\circ} with its transverse axis is$ (1) $x - y + 2 = 0$ (2) $x - y + 4 = 0$ (3) $x + y + 2 = 0$ (4) $x + y + 4 = 0$ 85. $Let the foci of a hyperbola coincide with the foci of the ellipse \frac{x ^{2}}{36} + \frac{y ^{2}}{16} = 1 . If the eccentricity of the hyperbola is 5, then the length of its latus rectum is:$ (1) $16$ (2) $245$ (3) $12$ (4) $\frac{96}{5}$ 86. $If A = (1, 2), B = (2, 1) and P is any point satisfying the condition PA + PB = 3, then the equation of the locus of P is$ (1) $16 x^{2} + 7 y^{2} - 64 x - 48 = 0$ (2) $x^{2} + 10 xy + 25 y^{2} - 34 x - 170 y = 0$ (3) $32 x^{2} + 8 xy + 32 y^{2} - 108 x - 108 y + 99 = 0$ (4) $4 x^{2} + 12 xy + 9 y^{2} - 20 x - 30 y = 0$ 87. $The locus of the mid-points of the chords of the circle x^{2} + y^{2} = 16 which are the tangents to the hyperbola 9 x^{2} - 16 y^{2} = 144 is$ (1) $3 x^{2} - 4 y^{2} = 16 (x^{2} + y^{2})$ (2) $4 x^{2} - 3 y^{2} = 9 (x^{2} + y^{2})$ (3) $16 x^{2} - 9 y^{2} = (x^{2} + y^{2})^{2}$ (4) $16 x^{2} - 9 y^{2} = 4 (x^{2} + y^{2})$ 88. $The angle between the tangents drawn to the parabola y^{2} = 12 x from the point (- 3, 2) is$ (1) $30^{\circ}$ (2) $45^{\circ}$ (3) $90^{\circ}$ (4) $60^{\circ}$ 89. $Let x + y = 0 be the radical axis of the circles S \equiv x^{2} + y^{2} + 2 gx + 2 fy + c = 0 and S^{1} \equiv x^{2} + y^{2} - 6 x - 4 y + 4 = 0 and the radius of the circle S = 0 be 1 . The g + f =$ (1) $\pm 5$ (2) $\pm 3$ (3) $\pm 2$ (4) $\pm 1$ 90. $The length of the latus rectum and directrices of a hyperbola with eccentricity e are 9 and x = \pm \frac{4}{13}, respectively. Let the line y - 3 x + 3 = 0 touch this hyperbola at (x_{0}, y_{0}) . If m is the product of the focal distances of the point (x_{0}, y_{0}), then 4 e^{2} + m is equal to_____$