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Yukti Vault

Coordinate Geometry

Hyperbola: JEE Main & Advanced PYQs

112 previous year questions on Hyperbola from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 51-100 of 112

  1. Let the product of the focal distances of the point $P(4,2\sqrt{3})$ on the hyperbola $H:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ be $32$. Let the length of the …
  2. If the eccentricity $e$ of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$, passing through $(6,4\sqrt3)$, satisfies $15(e^{2}+1)=34e$, then the length of the …
  3. Let $P(3\sec\theta,2\tan\theta)$ and $Q(3\sec\phi,2\tan\phi)$ where $\theta+\phi=\frac{\pi}{2}$, be two distinct points on the hyperbola $\frac{x^2}{9}-\frac{y^2}{4}=1$. …
  4. Let $e_1$ and $e_2$ be two distinct roots of the equation $x^{2}-ax+2=0$. Let the sets …
  5. For the Hyperbola $\frac{x^{2}}{\cos^{2}\alpha}-\frac{y^{2}}{\sin^{2}\alpha}=1$, which of the following remains constant when $\alpha$ varies ?
  6. A common tangent to the conics $x^{2}=6y$ and $2x^{2}-4y^{2}=9$ is :
  7. A tangent to the hyperbola $\frac{x^{2}}{4}-\frac{y^{2}}{2}=1$ meets $x$-axis at $P$ and $y$-axis at $Q$. Lines $PR$ and $QR$ are drawn such that $OPRQ$ is a rectangle …
  8. The locus of the point of intersection of the straight lines $tx-2y-3t=0$ and $x-2ty+3=0$ ($t\in R$), is :
  9. Tangents are drawn to the hyperbola $4x^2-y^2=36$ at the points $P$ and $Q$. If these tangents intersect at the point $T(0,3)$ then the area (in sq. units) of …
  10. The tangent at an extremity (in the first quadrant) of the latus rectum of the hyperbola $\frac{x^{2}}{4}-\frac{y^{2}}{5}=1$, meets the $x$-axis and $y$-axis at $A$ and …
  11. A hyperbola passes through the point $P(\sqrt2,\sqrt3)$ and has foci at $(\pm2,0)$. Then the tangent to this hyperbola at $P$ also passes through the point :
  12. If the eccentricity of the standard hyperbola passing through the point $(4,6)$ is $2$, then the equation of the tangent to the hyperbola at $(4,6)$ is :
  13. If the line $y=mx+7\sqrt3$ is normal to the hyperbola $\frac{x^{2}}{24}-\frac{y^{2}}{18}=1$, then a value of $m$ is :
  14. If a hyperbola has length of its conjugate axis equal to $5$ and the distance between its foci is $13$, then the eccentricity of the hyperbola is :
  15. The equation of a tangent to the hyperbola $4x^{2}-5y^{2}=20$, parallel to the line $x-y=2$, is
  16. A hyperbola whose transverse axis is along the major axis of the conic $\frac{x^{2}}{3}+\frac{y^{2}}{4}=4$ and has vertices at the foci of the conic. If the eccentricity …
  17. Let $H$ be the hyperbola, whose foci are $\left(1\pm\sqrt2,0\right)$ and eccentricity is $\sqrt2$. Then the length of its latus rectum is :
  18. The equation $\dfrac{x^{2}}{1-r}-\dfrac{y^{2}}{1+r}=1$, $r>1$, represents
  19. The vertices of a hyperbola H are $(\pm6,0)$ and its eccentricity is $\frac{\sqrt5}{2}$. Let N be the normal to H at a point in the first quadrant and parallel to the …
  20. A normal to the hyperbola $4x^{2}-9y^{2}=36$ meets the co-ordinate axes $x$ and $y$ at $A$ and $B$, respectively. If the parallelogram $OABP$ ($O$ being the origin) is …
  21. Let $P(x_0,y_0)$ be the point on the hyperbola $3x^{2}-4y^{2}=36$, which is nearest to the line $3x+2y=1$. Then $\sqrt2(y_0-x_0)$ is equal to :
  22. If $5x+9=0$ is the directrix of the hyperbola $16x^{2}-9y^{2}=144$, then its corresponding focus is :
  23. Let $S$ denote the locus of the point of intersection of the pair of lines $$4x-3y=12\alpha,$$ $$4\alpha x+3\alpha y=12,$$ where $\alpha$ varies over the set of non-zero …
  24. Let $0<\theta<\frac\pi2$. If the eccentricity of the hyperbola $\frac{x^2}{\cos^2\theta}-\frac{y^2}{\sin^2\theta}=1$ is greater than $2$, then the length of its latus …
  25. Consider the ellipse $E$ given by $\frac{x^2}{18}+\frac{y^2}{12}=1$. Let $H$ be the hyperbola whose eccentricity is the reciprocal of the eccentricity of $E$ and whose …
  26. Given the points $A(0,4)$ and $B(0,-4)$, the equation of the locus of the point $P(x,y)$ such that $|AP-BP|=6$ is ______.
  27. The locus of the point of intersection of the lines $\sqrt2x-y+4\sqrt2k=0$ and $\sqrt2kx+ky-4\sqrt2=0$ ($k$ is any non-zero real parameter) is
  28. If the tangents drawn to the hyperbola $4y^{2}=x^{2}+1$ intersect the co-ordinate axes at the distinct points $A$ and $B$, then the locus of the mid point of $AB$ is
  29. If a directrix of a hyperbola centered at the origin and passing through the point $\left(4,-2\sqrt3\right)$ is $5x=4\sqrt5$ and its eccentricity is $e$, then :
  30. Let $a$ and $b$ respectively be the semi-transverse and semi-conjugate axes of a standard hyperbola whose eccentricity satisfies the equation $9e^{2}-18e+5=0$. If …
  31. The eccentricity of the hyperbola whose length of its conjugate axis is equal to half of the distance between its foci, is
  32. If the vertices of a hyperbola be at $(-2,0)$ and $(2,0)$ and one of its foci be at $(-3,0)$, then which one of the following points does not lie on this hyperbola?
  33. Consider the hyperbola $$\frac{x^{2}}{100}-\frac{y^{2}}{64}=1$$ with foci at $S$ and $S_{1}$, where $S$ lies on the positive $x$-axis. Let $P$ be a point on the …
  34. A hyperbola has its centre at the origin, passes through the point $(4,2)$ and has transverse axis of length $4$ along the $x$-axis. Then the eccentricity of the …
  35. Let $P(a\sec\theta,b\tan\theta)$ and $Q(a\sec\phi,b\tan\phi)$, where $\theta+\phi=\pi/2$, be two points on the hyperbola $\dfrac{x^{2}}{a^{2}}-\dfrac{y^{2}}{b^{2}}=1$. …
  36. If $x=9$ is the chord of contact of the hyperbola $x^{2}-y^{2}=9$, then the equation of the corresponding pair of tangents is
  37. For hyperbola $\dfrac{x^2}{\cos^2\alpha}-\dfrac{y^2}{\sin^2\alpha}=1$ which of the following remains constant with change in $\alpha$
  38. If a hyperbola passes through the focus of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$ and its transverse and conjugate axes coincide with the major and minor axes …
  39. If the line $2x+\sqrt6y=2$ touches the hyperbola $x^{2}-2y^{2}=4$, then the point of contact is
  40. Tangents are drawn from any point on the hyperbola $\dfrac{x^{2}}{9}-\dfrac{y^{2}}{4}=1$ to the circle $x^{2}+y^{2}=9$. Find the locus of the mid-point of the chord of …
  41. Let $P(10,2\sqrt{15})$ be a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$, whose foci are $S$ and $S'$. If the length of its latus rectum is $8$, then the …
  42. Let the ellipse $E:\frac{x^2}{144}+\frac{y^2}{169}=1$ and the hyperbola $H:\frac{x^2}{16}-\frac{y^2}{\lambda^2}=-1$ have the same foci. If $e$ and $L$ respectively …
  43. For some $\theta\in\left(0,\frac\pi2\right)$, let the eccentricity and the length of the latus rectum of the hyperbola $x^2-y^2\sec^2\theta=8$ be $e_1$ and $l_1$ …
  44. If the line $\alpha x+2y=1$, where $\alpha\in\mathbf{R}$, does not meet the hyperbola $x^2-9y^2=9$, then a possible value of $\alpha$ is :
  45. 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 …
  46. Let $PQ$ be a chord of the hyperbola $\frac{x^2}{4}-\frac{y^2}{b^2}=1$, perpendicular to the $x$-axis such that $OPQ$ is an equilateral triangle, $O$ being the centre of …
  47. Let the domain of the function $f(x)=\log_3\log_5\log_7\left(9x-x^2-13\right)$ be the interval $(m,n)$. Let the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ have …
  48. A hyperbola, having the transverse axis of length $2\sin\theta$, is confocal with the ellipse $3x^2+4y^2=12$. Then its equation is
  49. Consider a branch of the hyperbola $$x^2-2y^2-2\sqrt{2}x-4\sqrt{2}y-6=0$$ with vertex at the point $A$. Let $B$ be one of the end points of its latus rectum. If $C$ is …
  50. Let $P(6,3)$ be a point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If the normal at the point $P$ intersects the $x$-axis at $(9,0)$, then the eccentricity of …

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