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Yukti Vault

Coordinate Geometry

Ellipse: JEE Main & Advanced PYQs

187 previous year questions on Ellipse 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 187

  1. Let $A$ be the point $(3,0)$ and circles with variable diameter $AB$ touch the circle $x^{2}+y^{2}=36$ internally. Let the curve $C$ be the locus of the point $B$. If …
  2. Let $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ ($a>b$) be a given ellipse, length of whose latus rectum is $10$. If its eccentricity is the maximum value of the function …
  3. Let the line $y=mx$ and the ellipse $2x^2+y^2=1$ intersect at a point $P$ in the first quadrant. If the normal to this ellipse at $P$ meets the co-ordinate axes at …
  4. A ray of light through $(2,1)$ is reflected at a point $P$ on the $y$-axis and then passes through the point $(5,3)$. If this reflected ray is the directrix of an …
  5. Let the common tangents to the curves $4\left(x^{2}+y^{2}\right)=9$ and $y^{2}=4x$ intersect at the point $Q$. Let an ellipse, centered at the origin $O$, has lengths of …
  6. Let $E$ be an ellipse whose axes are parallel to the co-ordinates axes, having its center at $(3,-4)$, one focus at $(4,-4)$ and one vertex at $(5,-4)$. If $mx-y=4$ …
  7. The acute angle between the pair of tangents drawn to the ellipse $2x^2+3y^2=5$ from the point $(1,3)$ is
  8. Let the eccentricity of an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ is reciprocal to that of the hyperbola $2x^{2}-2y^{2}=1$. If the ellipse intersects the …
  9. Let an ellipse with centre $(1,0)$ and latus rectum of length $\frac{1}{2}$ have its major axis along $x$-axis. If its minor axis subtends an angle $60^{\circ}$ at the …
  10. The length of the minor axis (along $y$-axis) of an ellipse in the standard form is $\frac4{\sqrt3}$. If this ellipse touches the line, $x+6y=8$; then its eccentricity …
  11. If the minimum area of the triangle formed by a tangent to the ellipse $\frac{x^{2}}{b^{2}}+\frac{y^{2}}{4a^{2}}=1$ and the co-ordinate axis is $kab$, then $k$ is equal …
  12. Let $x=4$ be a directrix to an ellipse whose centre is at the origin and its eccentricity is $\frac12$. If $P(1,\beta)$, $\beta>0$ is a point on this ellipse, then the …
  13. For real numbers $a,b$ ($a>b>0$), let $\text{Area}\left\{(x,y):x^{2}+y^{2}\le a^{2}\text{ and }\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}\ge1\right\}=30\pi$ and …
  14. If the length of the latus rectum of the ellipse $x^2+4y^2+2x+8y-\lambda=0$ is $4$, and $l$ is the length of its major axis, then $\lambda+l$ is equal to ________.
  15. An ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through the vertices of the hyperbola $H:\frac{x^2}{49}-\frac{y^2}{64}=-1$. Let the major and minor axes of the …
  16. If the points of intersection of two distinct conics $x^{2}+y^{2}=4b$ and $\frac{x^{2}}{16}+\frac{y^{2}}{b^{2}}=1$ lie on the curve $y^{2}=3x^{2}$, then $3\sqrt{3}$ …
  17. For some $\theta\in\left(0,\frac\pi2\right)$, if the eccentricity of the hyperbola, $x^2-y^2\sec^2\theta=10$ is $\sqrt5$ times the eccentricity of the ellipse …
  18. Let $S=\{(x,y)\in\mathbb{N}\times\mathbb{N}:9(x-3)^2+16(y-4)^2\le144\}$ and $T=\{(x,y)\in\mathbb{R}\times\mathbb{R}:(x-7)^2+(y-4)^2\le36\}$. Then $n(S\cap T)$ is equal …
  19. Let $A\left(\frac{3}{\sqrt{a}},\sqrt{a}\right)$, $a>0$, be a fixed point in the $xy$-plane. The image of $A$ in $y$-axis be $B$ and the image of $B$ in $x$-axis be $C$. …
  20. A common tangent $T$ to the curves $C_1:\frac{x^2}{4}+\frac{y^2}{9}=1$ and $C_2:\frac{x^2}{42}-\frac{y^2}{143}=1$ does not pass through the fourth quadrant. If $T$ …
  21. Let the tangents at the points $P$ and $Q$ on the ellipse $\frac{x^2}{2}+\frac{y^2}{4}=1$ meet at the point $R(\sqrt2,2\sqrt2-2)$. If $S$ is the focus of the ellipse on …
  22. Let a line $L$ pass through the point of intersection of the lines $bx+10y-8=0$ and $2x-3y=0$, $b\in\mathbf{R}-\left\{\frac43\right\}$. If the line $L$ also passes …
  23. Let $L$ be a common tangent line to the curves $4x^{2}+9y^{2}=36$ and $(2x)^{2}+(2y)^{2}=31$. Then the square of the slope of the line $L$ is $\_\_\_\_\_\_$ .
  24. Let $C$ be the circle $x^{2}+(y-1)^{2}=2$, $E_1$ and $E_2$ be two ellipses whose centres lie at the origin and major axes lie on $x$-axis and $y$-axis respectively. Let …
  25. If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity $e$ of the ellipse satisfies :
  26. If the co-ordinates of two points $A$ and $B$ are $(\sqrt7,0)$ and $(-\sqrt7,0)$ respectively and $P$ is any point on the conic, $9x^2+16y^2=144$, then $PA+PB$ is equal …
  27. If $e_1$ and $e_2$ are the eccentricities of the ellipse, $\frac{x^2}{18}+\frac{y^2}4=1$ and the hyperbola, $\frac{x^2}9-\frac{y^2}4=1$ respectively and $(e_1,e_2)$ is a …
  28. Let $P(x_1,y_1)$ and $Q(x_2,y_2)$ be two distinct points on the ellipse $$\frac{x^2}{9}+\frac{y^2}{4}=1$$ such that $y_1>0$, and $y_2>0$. Let $C$ denote the circle …
  29. Consider the ellipse $$\frac{x^2}{4}+\frac{y^2}{3}=1.$$ Let $H(\alpha,0)$, $0<\alpha<2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis crosses …
  30. Let $e_1$ be the eccentricity of the hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$ and $e_2$ be the eccentricity of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $a>b$ …
  31. If the distance between the foci of an ellipse is half the length of its latus rectum, then the eccentricity of the ellipse is :
  32. The tangent and normal to the ellipse $3x^{2}+5y^{2}=32$ at the point $P(2,2)$ meet the $x$-axis at $Q$ and $R$, respectively. Then the area (in sq. units) of the …
  33. If $\alpha x+\beta y=109$ is the equation of the chord of the ellipse $\frac{x^{2}}9+\frac{y^{2}}4=1$, whose mid point is $\left(\frac52,\frac12\right)$, then …
  34. Define the collections $\{E_1,E_2,E_3,\dots\}$ of ellipses and $\{R_1,R_2,R_3,\dots\}$ of rectangles as follows:\\ $E_1:\frac{x^2}{9}+\frac{y^2}{4}=1$;\\ $R_1$ …
  35. Let the ellipse $E_1:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $a>b$ and $E_2:\frac{x^2}{A^2}+\frac{y^2}{B^2}=1$, $A<B$ have same eccentricity $\frac1{\sqrt3}$. Let the …
  36. Consider the parabola $P: y^{2}=4kx$ and the ellipse $E:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$. Let the line segment joining the points of intersection of $P$ and …
  37. If the maximum distance of normal to the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{b^{2}}=1$, $b<2$, from the origin is $1$, then the eccentricity of the ellipse is:
  38. Consider the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$. Let $S(p,q)$ be a point in the first quadrant such that $\frac{p^2}{9}+\frac{q^2}{4}>1$. Two tangents are drawn …
  39. Let $S$ and $S'$ be the foci of an ellipse and $B$ be any one of the extremities of its minor axis. If $\triangle S'BS$ is a right angled triangle with right angle at …
  40. An ellipse, with foci at $(0,2)$ and $(0,-2)$ and minor axis of length $4$, passes through which of the following points?
  41. If the tangent at a point on the ellipse $\frac{x^{2}}{27}+\frac{y^{2}}{3}=1$ meets the coordinate axes at $A$ and $B$, and $O$ is the origin, then the minimum area (in …
  42. Let a tangent to the curve $9x^{2}+16y^{2}=144$ intersect the coordinate axes at the points $A$ and $B$. Then, the minimum length of the line segment $AB$ is ______
  43. If the length of the latus rectum of an ellipse is $4$ units and the distance between a focus and its nearest vertex on the major axis is $\frac32$ units, then its …
  44. If $\beta$ is one of the angles between the normals to the ellipse, $x^{2}+3y^{2}=9$ at the points $(3\cos\theta,\sqrt3\sin\theta)$ and …
  45. If the normal to the ellipse $3x^2+4y^2=12$ at a point $P$ on it is parallel to the line, $2x+y=4$ and the tangent to the ellipse at $P$ passes through $Q(4,4)$ then …
  46. Let $C$ be the largest circle centred at $(2,0)$ and inscribed in the ellipse $\frac{x^{2}}{36}+\frac{y^{2}}{16}=1$. If $(1,\alpha)$ lies on $C$, then $10\alpha^{2}$ is …
  47. If the line $x-2y=12$ is a tangent to the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ at the point $\left(3,-\frac92\right)$, then the length of the latus rectum …
  48. An ellipse passes through the foci of the hyperbola $9x^2-4y^2=36$ and its major and minor axes lie along the transverse and conjugate axes of the hyperbola …
  49. The line $x=8$ is the directrix of the ellipse $E:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ with the corresponding focus $(2,0)$. If the tangent to $E$ at the point $P$ …
  50. Let the product of the focal distances of the point $\left(\sqrt3,\frac12\right)$ on the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, $(a>b)$, be $\frac74$. Then …

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