Hyperbola PYQs Set 1 (PDF)

Set 1 of 11 · Questions 1-30 of 302

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Mere Teacher · Previous Year Questions JEE Main Mathematics · Hyperbola · Set 1 (Q1-30)
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1. $The equation of the hyperbola whose foci are (-2, 0) and (2, 0) and eccentricity is 2 is given by$ (1) $x2 - 3y2 = 3$ (2) $3x2 - y2 = 3$ (3) $- x2 + 3y2 = 3$ (4) $- 3x2 + y2 = 3$ 2. $The length of the chord of the parabola x^{2} = 4 y havingequation x - 2 y + 42 = 0 is:$ (1) $32$ (2) $211$ (3) $82$ (4) $63$ 3. $The equation of the pair of asymptotes of the hyperbola x^{2} - 2 y^{2} - 8 x + 8 y + 4 = 0 is$ (1) $x^{2} - 2 y^{2} - 8 x + 8 y - 8 = 0$ (2) $2 x^{2} - 4 y^{2} - 16 x + 16 y - 7 = 0$ (3) $x^{2} - 2 y^{2} - 8 x + 8 y + 8 = 0$ (4) $2 x^{2} - 4 y^{2} - 16 x + 16 y + 9 = 0$ 4. $Suppose that the circle x^{2} + y^{2} + 2 gx + 2 fy + c = 0 has its centre on 2 x + 3 y - 7 = 0 and cuts the circles x^{2} + y^{2} - 4 x - 6 y + 11 = 0 and x^{2} + y^{2} - 10 x - 4 y + 21 = 0 orthogonally. Then 5 g - 10 f + 3 c =$ (1) $0$ (2) $1$ (3) $3$ (4) $9$ 5. $The product of lengths of perpendicular from any point on the hyperbola x^{2} - y^{2} = 16 to its asymptotes, is$ (1) $2$ (2) $4$ (3) $8$ (4) $16$ 6. $If x^{2} = 8 ay is the transformed equation of x^{2} - 4 y + 6 x + 15 = 0 when the origin is shifted to the point (α, β) by translation of axes, then 2 α + 8 β^{2} =$ (1) $8$ (2) $18$ (3) $12$ (4) $16$ 7. $The point where the line 4 x - 3 y + 7 = 0 touches the circle x^{2} + y^{2} - 6 x + 4 y - 12 = 0 is$ (1) $(1, 1)$ (2) $(1, - 1)$ (3) $(- 1, 1)$ (4) $(- 1, - 1)$ 8. $When the coordinate axes are rotated about the origin in the positive direction through an angle \frac{π}{4}, if the equation 49 x^{2} + 25 y^{2} = 1225 is transformed to p x^{2} + qxy + r y^{2} = t and the G.C.D of p, q, r, t is 1 , then$ (1) $(p - q + r - 32)^{2} = 4 t$ (2) $(p - q - r + 12)^{2} = t$ (3) $(p + q + r - 15)^{2} = t$ (4) $(- p - q + r + 13)^{2} = t$ 9. $Let L be a normal to the parabola y2 = 4x. If L passes through the point (9, 6), then L is given by$ (1) $y - x + 3 = 0$ (2) $y + 3x - 33 = 0$ (3) $y + x - 15 = 0$ (4) $y - 2x + 12 = 0$ 10. $The tangent to the curve xy = 25 at any point on it cuts the coordinate axes at A and B, then the area of the triangle OAB is$ (1) $100 sq. units$ (2) $50 sq. units$ (3) $25 sq. units$ (4) $75 sq. units$ 11. $The centre of the circle passing through the points of intersection of the circles (x + 3)^{2} + (y + 2)^{2} = 25 and (x - 2)^{2} + (y - 3)^{2} = 25 and cutting the circle (x + 1)^{2} + (y - 2)^{2} = 16 orthogonally is$ (1) $(\frac{- 27}{2}, \frac{- 25}{2})$ (2) $(0, 0)$ (3) $(\frac{16}{3}, \frac{- 25}{4})$ (4) $(\frac{4}{7}, \frac{3}{7})$ 12. $The radius of the circle whose center lies at (1, 2) while cutting the circle x^{2} + y^{2} + 4 x + 16 y - 30 = 0 orthogonally, is .......... units.$ (1) $41$ (2) $31$ (3) $21$ (4) $11$ 13. $The values of m for which the line y = mx + 2 becomes a tangent to the hyperbola 4 x^{2} - 9 y^{2} = 36 is$ (1) $\pm \frac{2}{3}$ (2) $\pm \frac{2 2}{3}$ (3) $\pm \frac{8}{9}$ (4) $\pm \frac{4 2}{3}$ 14. $x = 4 (1 + cos θ) and y = 3 (1 + sin θ) are the parametric equations of$ (1) $\frac{( x - 3 ) ^{2}}{9} + \frac{( y - 4 ) ^{2}}{16} = 1$ (2) $\frac{( x + 4 ) ^{2}}{16} + \frac{( y + 3 ) ^{2}}{9} = 1$ (3) $\frac{( x - 4 ) ^{2}}{16} - \frac{( y - 3 ) ^{2}}{9} = 1$ (4) $\frac{( x - 4 ) ^{2}}{16} + \frac{( y - 3 ) ^{2}}{9} = 1$ 15. $If A and B are the points of intersection of the circle x^{2} + y^{2} - 8 x = 0 and the hyperbola \frac{x ^{2}}{9} - \frac{y ^{2}}{4} = 1 and a point P moves on the line 2 x - 3 y + 4 = 0, then the centroid of △ PAB lies on the line:$ (1) $x + 9 y = 36$ (2) $4 x - 9 y = 12$ (3) $6 x - 9 y = 20$ (4) $9 x - 9 y = 32$ 16. $If e_{1} and e_{2} are the eccentricities of a hyperbola 3 x^{2} - 3 y^{2} = 25 and its conjugate, then$ (1) $e_{1} + e_{2} = 4$ (2) $e_{1} + e_{2} = 2$ (3) $e_{1}^{2} + e_{2}^{2} = 2$ (4) $e_{1}^{2} + e_{2}^{2} = 4$ 17. $Area of a triangle formed by tangent and normal to the curve \frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1 at p (\frac{a}{2}, \frac{b}{2}) with the X -axis is$ (1) $4 ab$ (2) $\frac{ab a ^{2} + b ^{2}}{4}$ (3) $\frac{ab a ^{2} - b ^{2}}{4}$ (4) $\frac{b ( a ^{2} + b ^{2} )}{4 a}$ 18. $The equation of the circle whose radius is 3 and which touches internally the circle x^{2} + y^{2} - 4 x - 6 y - 12 = 0 at the point (- 1, - 1) is$ (1) $5 x^{2} + 5 y^{2} + 9 x - 6 y - 7 = 0$ (2) $5 x^{2} + 5 y^{2} - 8 x - 14 y - 32 = 0$ (3) $5 x^{2} + 5 y^{2} - 6 x + 8 y - 8 = 0$ (4) $5 x^{2} + 5 y^{2} + 6 x - 8 y - 12 = 0$ 19. $If (a, b) is the centre of the circle passing through the vertices of the triangle formed by x + y = 6, 2 x + y = 4 and x + 2 y = 5, then (a, b) is$ (1) $(- 17, - 16)$ (2) $(\frac{17}{2}, \frac{19}{2})$ (3) $(17, 18)$ (4) $(\frac{- 17}{2}, \frac{- 19}{2})$ 20. $The distance between the foci of a hyperbola is 16 and its eccentricity is 2 . Its equation is$ (1) $2 x^{2} - 3 y^{2} = 7$ (2) $x^{2} - y^{2} = 32$ (3) $y^{2} - x^{2} = 32$ (4) $\frac{x ^{2}}{4} - \frac{y ^{2}}{9} = 1$ 21. $The equation of the circle whose diameter is the common chord of the circles x^{2} + y^{2} + 2 x + 2 y + 1 = 0 and x^{2} + y^{2} + 4 x + 6 y + 4 = 0 is$ (1) $10 x^{2} + 10 y^{2} + 14 x + 8 y + 1 = 0$ (2) $3 x^{2} + 3 y^{2} - 3 x + 6 y - 8 = 0$ (3) $2 x^{2} + 2 y^{2} - 2 x + 4 y + 1 = 0$ (4) $x^{2} + y^{2} - x + 2 y + 4 = 0$ 22. $Let e_{1} and e_{2} be the eccentricities of the ellipse \frac{x ^{2}}{b ^{2}} + \frac{y ^{2}}{25} = 1 and the hyperbola \frac{x ^{2}}{16} - \frac{y ^{2}}{b ^{2}} = 1, respectively. If b < 5 and e_{1} e_{2} = 1, then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is:$ (1) $\frac{4}{5}$ (2) $\frac{3}{5}$ (3) $\frac{7}{4}$ (4) $\frac{3}{2}$ 23. $The value of a, such that the power of the point (1, 6) with respect to the circle x^{2} + y^{2} + 4 x - 6 y - a = 0, is -16 , is$ (1) $7$ (2) $11$ (3) $13$ (4) $21$ 24. $A hyperbola, having the transverse axis of length 2 sin θ, is confocal with the ellipse 3 x^{2} + 4 y^{2} = 12. Then its equation is$ (1) $x2 cosec2θ - y2 sec2θ = 1$ (2) $x2 sec2θ - y2 cosec2θ = 1$ (3) $x2 sin2θ - y2 cos2θ = 1$ (4) $x2 cos2 θ - y2 sin2θ = 1$ 25. $A (- 2, 9) and B (1, 6) are two points on the curve y = x^{2} + 5 . The coordinates of the point C on the curve such that the tangent drawn at A is parallel to the chord BC, is$ (1) $(- 5, 30)$ (2) $(0, 5)$ (3) $(- 9, 86)$ (4) $(6, 41)$ 26. $The length of the common chord of the circles x^{2} + y^{2} + 3 x + 5 y + 4 = 0 and x^{2} + y^{2} + 5 x + 3 y + 4 = 0 is _____ units.$ (1) $3$ (2) $2$ (3) $6$ (4) $4$ 27. $The length of latus rectum of the parabola 4 y^{2} + 3 x + 3 y + 1 = 0 is$ (1) $\frac{4}{3}$ (2) $7$ (3) $12$ (4) $\frac{3}{4}$ 28. $Which circle among the following bisects the circumference of the circle x^{2} + y^{2} - 8 x - 6 y + 23 = 0 ?$ (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) $x^{2} + y^{2} + 6 x - 4 y + 9 = 0$ 29. $If (1, 2) is the focus, x + 2 y = 0 is the directrix and 2 is the eccentricity of a hyperbola then the equation of the hyperbola is$ (1) $x^{2} - y^{2} = a^{2}$ (2) $3 x^{2} - 8 xy - 3 y^{2} - 10 x - 20 y + 25 = 0$ (3) $xy = c^{2}$ (4) $3 x^{2} - 8 xy - 3 y^{2} + 10 x - 20 y - 25 = 0$ 30. $Let A be a square matrix of order 2 such that \| A \| = 2 and the sum of its diagonal elements is -3. If the points (x, y) satisfying A^{2} + x A + y I = O lie on a hyperbola, whose length of semi major axis is x and semi minor axis is y, eccentricity is e and the length of the latus rectum is l, then 81 (e^{4} + l^{2}) is equal to$