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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 1-50 of 112

  1. Let the foci of a hyperbola $H$ coincide with the foci of the ellipse $E:\frac{(x-1)^{2}}{100}+\frac{(y-1)^{2}}{75}=1$ and the eccentricity of the hyperbola $H$ be the …
  2. Let the foci of the ellipse $\frac{x^2}{16}+\frac{y^2}{7}=1$ and the hyperbola $\frac{x^2}{144}-\frac{y^2}{\alpha}=\frac{1}{25}$ coincide. Then the length of the latus …
  3. The normal to the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{9}=1$ at the point $(8,3\sqrt{3})$ on it passes through the point :
  4. Let the eccentricity $e$ of a hyperbola satisfy the equation $6e^{2}-11e+3=0$. If the foci of the hyperbola are $(3,5)$ and $(3,-4)$, then the length of its latus rectum …
  5. Let $P$ be a point on the hyperbola $H:\frac{x^{2}}{9}-\frac{y^{2}}{4}=1$, in the first quadrant such that the area of triangle formed by $P$ and the two foci of $H$ is …
  6. If the line $\alpha x+\beta y=1$ touches the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$, then the locus of the point $(\alpha,\beta)$ is
  7. If the foci of a hyperbola are same as that of the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{25}=1$ and the eccentricity of the hyperbola is $\frac{15}{8}$ times the …
  8. 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 $\frac{x^{2}}{16}-\frac{y^{2}}{b^{2}}=1$ …
  9. Let $\lambda x-2y=\mu$ be a tangent to the hyperbola $a^{2}x^{2}-y^{2}=b^{2}$. Then $\left(\frac{\lambda}{a}\right)^{2}-\left(\frac{\mu}{b}\right)^{2}$ is equal to :
  10. Let the sum of the focal distances of the point $P(4,3)$ on the hyperbola $H:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ be $8\sqrt{\frac{5}{3}}$. If for $H$, the length …
  11. For $0<\theta<\pi/2$, if the eccentricity of the hyperbola $x^{2}-y^{2}\operatorname{cosec}^{2}\theta=5$ is $\sqrt{7}$ times eccentricity of the ellipse …
  12. The point $P(-2\sqrt{6},\sqrt{3})$ lies on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ having eccentricity $\frac{\sqrt{5}}{2}$. If the tangent and normal …
  13. Consider a hyperbola $H:x^{2}-2y^{2}=4$. Let the tangent at a point $P(4,\sqrt{6})$ meet the $x$-axis at $Q$ and latus rectum at $R(x_1,y_1)$, $x_1>0$. If $F$ is a focus …
  14. Let $H:\frac{-x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ be the hyperbola, whose eccentricity is $\sqrt{3}$ and the length of the latus rectum is $4\sqrt{3}$. Suppose the …
  15. Let $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, $a>b$ be an ellipse, whose eccentricity is $\frac{1}{\sqrt{2}}$ and the length of the latusrectum is $\sqrt{14}$. Then …
  16. Let $a>0$, $b>0$. Let $e$ and $l$ respectively be the eccentricity and length of the latus rectum of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$. Let $e'$ …
  17. A hyperbola passes through the foci of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$ and its transverse and conjugate axes coincide with major and minor axes of the …
  18. The locus of the centroid of the triangle formed by any point $P$ on the hyperbola $16x^{2}-9y^{2}+32x+36y-164=0$, and its foci is :
  19. Let $H:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ be a hyperbola such that the distance between its foci is $6$ and the distance between its directrices is …
  20. Let $O$ be the origin, and $P$ and $Q$ be two points on the rectangular hyperbola $xy=12$ such that the mid point of the line segment $PQ$ is …
  21. Let one focus of the hyperbola $H:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ be at $(\sqrt{10},0)$ and the corresponding directrix be $x=\frac{9}{\sqrt{10}}$. If $e$ and …
  22. The locus of the mid points of the chords of the hyperbola $x^{2}-y^{2}=4$, which touch the parabola $y^{2}=8x$, is :
  23. Let the eccentricity of the hyperbola $H:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ be $\sqrt{\frac{5}{2}}$ and length of its latus rectum be $6\sqrt{2}$. If $y=2x+c$ is …
  24. A line parallel to the straight line $2x-y=0$ is tangent to the hyperbola $\frac{x^2}4-\frac{y^2}2=1$ at the point $(x_1,y_1)$. Then $x_1^2+5y_1^2$ is equal to :
  25. A square $ABCD$ has all its vertices on the curve $x^{2}y^{2}=1$. The midpoints of its sides also lie on the same curve. Then, the square of area of $ABCD$ is …
  26. Let the lengths of the transverse and conjugate axes of a hyperbola in standard form be $2a$ and $2b$, respectively, and one focus and the corresponding directrix of …
  27. Let the tangent to the parabola $y^{2}=12x$ at the point $(3,\alpha)$ be perpendicular to the line $2x+2y=3$. Then the square of distance of the point $(6,-4)$ from the …
  28. Let the equation of two diameters of a circle $x^2+y^2-2x+2fy+1=0$ be $2px-y=1$ and $2x+py=4p$. Then the slope $m\in(0,\infty)$ of the tangent to the hyperbola …
  29. Let the focal chord of the parabola $P:y^2=4x$ along the line $L:y=mx+c$, $m>0$ meet the parabola at the points $M$ and $N$. Let the line $L$ be a tangent to the …
  30. Let $H:\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$, $a>0$, $b>0$, be a hyperbola such that the sum of lengths of the transverse and the conjugate axes is …
  31. If a hyperbola passes through the point $P(10,16)$ and it has vertices at $(\pm6,0)$, then the equation of the normal to it at $P$ is :
  32. If the equation of the hyperbola with foci $(4,2)$ and $(8,2)$ is $3x^{2}-y^{2}-\alpha x+\beta y+\gamma=0$, then $\alpha+\beta+\gamma$ is equal to $\_\_\_\_\_\_$ .
  33. If foci of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{12}=1$ and the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ coincide, then the eccentricity of the hyperbola, with foci …
  34. Let $S$ be the focus of the hyperbola $\frac{x^{2}}{3}-\frac{y^{2}}{5}=1$, on the positive $x$-axis. Let $C$ be the circle with its centre at $A(\sqrt{6},\sqrt{5})$ and …
  35. The foci of a hyperbola are $(\pm2,0)$ and its eccentricity is $\frac{3}{2}$. A tangent, perpendicular to the line $2x+3y=6$, is drawn at a point in the first quadrant …
  36. Let $m_1$ and $m_2$ be the slopes of the tangents drawn from the point $P(4,1)$ to the hyperbola $H:\frac{y^{2}}{25}-\frac{x^{2}}{16}=1$. If $Q$ is the point from which …
  37. Let the foci and length of the latus rectum of an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, $a>b$ be $(\pm5,0)$ and $\sqrt{50}$, respectively. Then, the …
  38. Let $H_n:\frac{x^{2}}{1+n}-\frac{y^{2}}{3+n}=1$, $n\in\mathbb{N}$. Let $k$ be the smallest even value of $n$ such that the eccentricity of $H_k$ is a rational number. If …
  39. A hyperbola having the transverse axis of length $\sqrt2$ has the same foci as that of the ellipse $3x^2+4y^2=12$, then this hyperbola does not pass through which of the …
  40. Let the latus ractum of the hyperbola $\frac{x^{2}}{9}-\frac{y^{2}}{b^{2}}=1$ subtend an angle of $\frac{\pi}{3}$ at the centre of the hyperbola. If $b^{2}$ is equal to …
  41. Let the eccentricity of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ be $\frac{5}{4}$. If the equation of the normal at the point …
  42. Consider the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ having one of its focus at $P(-3,0)$. If the latus rectum through its other focus subtends a right …
  43. For the hyperbola $H:x^2-y^2=1$ and the ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $a>b>0$, let the (1) eccentricity of $E$ be reciprocal of the eccentricity of $H$ …
  44. Let the hyperbola $H:\frac{x^{2}}{a^{2}}-y^{2}=1$ and the ellipse $E:3x^{2}+4y^{2}=12$ be such that the length of latus rectum of $H$ is equal to the length of latus …
  45. If the line $x-1=0$ is a directrix of the hyperbola $kx^2-y^2=6$, then the hyperbola passes through the point
  46. Let a line $L_1$ be tangent to the hyperbola $\frac{x^{2}}{16}-\frac{y^{2}}{4}=1$ and let $L_2$ be the line passing through the origin and perpendicular to $L_1$. If the …
  47. Let the hyperbola $H:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ pass through the point $(2\sqrt2,-2\sqrt2)$. A parabola is drawn whose focus is same as the focus of $H$ with …
  48. Let $P(3,3)$ be a point on the hyperbola, $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$. If the normal to it at $P$ intersects the $x$-axis at $(9,0)$ and $e$ is its eccentricity …
  49. The locus of the point of intersection of the lines $(\sqrt{3})kx+ky-4\sqrt{3}=0$ and $\sqrt{3}x-y-4(\sqrt{3})k=0$ is a conic, whose eccentricity is $\_\_\_\_\_\_$ .
  50. Let $e_1$ and $e_2$ be the eccentricities of the ellipse, $\frac{x^2}{25}+\frac{y^2}{b^2}=1$ ($b<5$) and the hyperbola, $\frac{x^2}{16}-\frac{y^2}{b^2}=1$ respectively …

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