Hyperbola PYQs Set 2 (PDF)

Set 2 of 11 · Questions 31-60 of 302

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Hyperbola · Set 2 (Q31-60)
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31. $Let L_{1} be a straight line passing through the origin and L_{2} be the straight line x + y = 1 . If the intercepts made by the circle x^{2} + y^{2} - x + 3 y = 0 on L_{1} and L_{2} are equal, then which of the following equations represent L_{1}$ (1) $x + y = 0 and x + 7 y = 0$ (2) $x - y = 0 and x + 7 y = 0$ (3) $x - 7 y = 0 and x + y = 0$ (4) $x - 7 y = 0 and x - y = 0$ 32. $If the line y = mx + 73 is normal to the hyperbola \frac{x ^{2}}{24} - \frac{y ^{2}}{18} = 1, then a value of m is:$ (1) $\frac{5}{2}$ (2) $\frac{15}{2}$ (3) $\frac{2}{5}$ (4) $\frac{3}{5}$ 33. $The equation of a circle touching the coordinate axes and the line 3 x - 4 y = 12 is$ (1) $x^{2} + y^{2} + 6 x + 6 y + 9 = 0$ (2) $x^{2} + y^{2} + 6 x + 6 y - 9 = 0$ (3) $x^{2} + y^{2} - 6 x - 6 y + 9 = 0$ (4) $x^{2} + y^{2} - 6 x - 6 y - 9 = 0$ 34. $If (4, 2) and (k, - 3) are conjugate points with respect to x^{2} + y^{2} - 5 x + 8 y + 6 = 0, then k equals$ (1) $\frac{28}{3}$ (2) $- \frac{28}{3}$ (3) $\frac{3}{28}$ (4) $- \frac{3}{28}$ 35. $The distance between the focii of a hyperbola is 16 and its eccentricity is 2 . Its equation is$ (1) $x^{2} - y^{2} = 32$ (2) $\frac{x ^{2}}{4} - \frac{y ^{2}}{9} = 1$ (3) $2 x^{2} - 3 y^{2} = 7$ (4) $y^{2} - x^{2} = 32$ 36. $The equation of the normal to the hyperbola \frac{x ^{2}}{16} - \frac{y ^{2}}{9} = 1 at (- 4, 0) is$ (1) $x = 1$ (2) $y = 0$ (3) $2 x - 3 y = 1$ (4) $x = 0$ 37. $Let E : \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1, a > b and H : \frac{x ^{2}}{A ^{2}} - \frac{y ^{2}}{B ^{2}} = 1 . Let the distance between the foci of E and the foci of H be 23 . If a - A = 2, and the ratio of the eccentricities of E and H is \frac{1}{3}, then the sum of the lengths of their latus rectums is equal to:$ (1) $10$ (2) $9$ (3) $8$ (4) $7$ 38. $The ellipse \frac{x ^{2}}{25} + \frac{y ^{2}}{16} = 1 and the hyperbola \frac{x ^{2}}{25} - \frac{y ^{2}}{16} = 1 have in common$ (1) $centre only$ (2) $centre, foci and directrices$ (3) $centre, foci and vertices$ (4) $centre and vertices only.$ 39. $The centre of a circle is (2, - 3) and the circumference is 10 π . Then its equation is$ (1) $x^{2} + y^{2} + 4 x + 6 y + 12 = 0$ (2) $x^{2} + y^{2} - 4 x + 6 y + 12 = 0$ (3) $x^{2} + y^{2} - 4 x + 6 y - 12 = 0$ (4) $x^{2} + y^{2} - 4 x - 6 y - 12 = 0$ 40. $The equation of the smallest circle passing through the points (2, 2) and (3, 3) is$ (1) $x^{2} + y^{2} + 5 x - 5 y + 12 = 0$ (2) $x^{2} + y^{2} - 5 x + 5 y - 12 = 0$ (3) $x^{2} + y^{2} + 5 x + 5 y + 12 = 0$ (4) $x^{2} + y^{2} - 5 x - 5 y + 12 = 0$ 41. $If one of the roots of the equation x^{2} - 5 x - 14 = 0 is the length of the semi conjugate axis of the hyperbola \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1 and the square of the other root is the semi-transverse axis then the focus of the hyperbola that lies on the positive x -axis is$ (1) $(5, 0)$ (2) $(65, 0)$ (3) $(7, 0)$ (4) $(74, 0)$ 42. $If A (5, - 4) and B (7, 6) are points in a plane, then the set of all points P (x, y) in the plane such that AP : PB = 2 : 3 is$ (1) $a circle$ (2) $a hyperbola$ (3) $an ellipse$ (4) $a parabola$ 43. $If the equation x + y + n = 0 represents a normal to the hyperbola \frac{x ^{2}}{6} - \frac{y ^{2}}{2} = 1, then n =$ (1) $\pm 3$ (2) $\pm 4$ (3) $\pm 2$ (4) $\pm 2$ 44. $The equation of the circle whose diameter is the common chord of the circles x^{2} + y^{2} - 3 x + y - 10 = 0 and x^{2} + y^{2} - x + 2 y - 20 = 0 is$ (1) $x^{2} + y^{2} - 3 x + 6 y + 15 = 0$ (2) $x^{2} + y^{2} - 6 x + 4 y + 10 = 0$ (3) $x^{2} + y^{2} - 9 x + 2 y + 20 = 0$ (4) $x^{2} + y^{2} - 9 x - 2 y + 20 = 0$ 45. $Each of the two orthogonal circles C_{1} and C_{2} passes through both the points (2, 0) and (- 2, 0) . If y = mx + c is a common tangent to these circles, then$ (1) $c^{2} = 4 (1 + 2 m^{2})$ (2) $c^{2} = 2 (1 + 2 m^{2})$ (3) $c^{2} = 1 + m^{2}$ (4) $c^{2} m^{2} = 4 (1 + m^{2})$ 46. $If \frac{x ^{2}}{k - \frac{5}{2}} + \frac{y ^{2}}{\frac{7}{3} - k} = 1 (k is a real number) represents a hyperbola, then the set of all values of k is$ (1) $(- \infty, \frac{7}{3}) \cup (\frac{5}{2}, \infty)$ (2) $(\frac{7}{3}, \frac{5}{2})$ (3) $(- 1, \frac{7}{3}) \cup (\frac{5}{2}, 1)$ (4) $R - (\frac{7}{3}, \frac{5}{2})$ 47. $If a variable circle S = 0 touches the line y = x and passes through the point (0, 0), then the fixed point that lies on the common chord of the circles x^{2} + y^{2} + 6 x + 8 y - 7 = 0 and S = 0 is$ (1) $(\frac{1}{2} \frac{1}{2})$ (2) $(- \frac{1}{2}, - \frac{1}{2})$ (3) $(\frac{1}{2}, - \frac{1}{2})$ (4) $(- \frac{1}{2}, \frac{1}{2})$ 48. $If one focus of a hyperbola is ( 3,0 ), the equation of its directrix is 4 x - 3 y - 3 = 0 and its eccentricity e = 5/4, then the coordinates of its vertex is$ (1) $(\frac{3}{5}, \frac{11}{5})$ (2) $(\frac{11}{5} \frac{3}{5})$ (3) $(\frac{7}{5}, \frac{4}{5})$ (4) $(\frac{4}{5}, \frac{7}{5})$ 49. $The equation of the direct common tangent of the circles x^{2} + y^{2} - 6 x - 4 y - 23 = 0 and x^{2} + y^{2} + 2 x + 2 y + 1 = 0 is$ (1) $6 x - 4 y + 1 = 0$ (2) $3 x - 4 y + 6 = 0$ (3) $4 x + 3 y + 12 = 0$ (4) $2 x - 4 y + 3 = 0$ 50. $The lines x cos α + y sin α = P, α \in R are chords of the hyperbola \frac{x ^{2}}{9} - \frac{y ^{2}}{36} = 1 and they subtend a right angle at the centre of the hyperbola. The locus of the poles of these lines with respect to the given hyperbola is$ (1) $x^{2} - 16 y^{2} = 108$ (2) $16 x^{2} - y^{2} = 108$ (3) $16 x^{2} + y^{2} = 108$ (4) $x^{2} + 16 y^{2} = 108$ 51. $If the equation of the circle having its centre in the second quadrant touches the coordinate axes and also the line \frac{x}{5} + \frac{y}{12} = 1 is x^{2} + y^{2} + 2 λx - 2 λy + λ^{2} = 0, then λ =$ (1) $3$ (2) $10$ (3) $15$ (4) $-2$ 52. $Let C be a curve a x^{2} + 2 hxy + b y^{2} + 2 gx + 2 fy + c = 0 in a cartesian plane. By rotating the coordinate axes through an angle \frac{π}{4} in the positive direction, if the transformed equation of C is Y^{2} + XY - X = 0, then (h^{2} - ab) - 2 gf =$ (1) $0$ (2) $2$ (3) $1$ (4) $-1$ 53. $A rectangular hyperbola passing through (3, 2) has its asymptotes parallel to the coordinate axes. If (1, 1) is the point of intersection of the two perpendicular tangents of that hyperbola, then its equation is$ (1) $xy = x + \frac{1}{y}$ (2) $x (y + 1 + \frac{1}{x}) = 1$ (3) $x (1 - y) = y - 1$ (4) $xy = x + y + 1$ 54. $The normal at a point on the parabola y^{2} = 4 x passes through (5, 0) . If there are two more normals to this parabola passing through (5, 0), then the equation of one of these normals is$ (1) $2 x - y - 10 = 0$ (2) $x + y - 5 = 0$ (3) $3 x + 2 y + 53 = 0$ (4) $3 x - y - 53 = 0$ 55. $The equation of the circle which cuts the circles S_{1} \equiv x^{2} + y^{2} - 4 = 0 S_{2} \equiv x^{2} + y^{2} - 6 x - 8 y + 10 = 0 S_{3} \equiv x^{2} + y^{2} + 2 x - 4 y - 2 = 0 at the extremities of diameters of these circles is$ (1) $x^{2} + y^{2} - 4 x - 6 y - 4 = 0$ (2) $x^{2} + y^{2} + 4 x - 4 = 0$ (3) $x^{2} + y^{2} = 25$ (4) $x^{2} + y^{2} + x + y + 1 = 0$ 56. $For 0 < θ < π /2, if the eccentricity of the hyperbola x^{2} - y^{2} cosec^{2} θ = 5 is 7 times eccentricity of the ellipse x^{2} cosec^{2} θ + y^{2} = 5, then the value of θ is :$ (1) $\frac{π}{6}$ (2) $\frac{5 π}{12}$ (3) $\frac{π}{3}$ (4) $\frac{π}{4}$ 57. $Find the equation of the circle passing through (1, - 2) and touching the X -axis at (3, 0) .$ (1) $x^{2} + y^{2} + 6 x - 4 y - 9 = 0$ (2) $x^{2} + y^{2} - 6 x - 4 y + 9 = 0$ (3) $x^{2} + y^{2} - 6 x - 4 y - 9 = 0$ (4) $x^{2} + y^{2} - 6 x + 4 y + 9 = 0$ 58. $Find the value of m + n, if the circumference of the circle x^{2} + y^{2} + 8 x + 8 y - m = 0 is bisected by the circle x^{2} + y^{2} - 2 x + 4 y + n = 0 .$ (1) $-56$ (2) $56$ (3) $50$ (4) $-34$ 59. $The locus of the point of intersection of the lines (3) kx + ky - 43 = 0 and 3 x - y - 4 (3) k = 0 is aconic, whose eccentricity is$ 60. $Let the product of the focal distances of the point P (4, 23) on the hyperbola H : \frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1 be 32. Let the length of the conjugate axis of H be p and the length of its latus rectum be q . Then p^{2} + q^{2} is equal to _______.$