Skip to main content
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.

Practice this chapter

Questions 1-50 of 187

  1. On the ellipse $\frac{x^{2}}{8}+\frac{y^{2}}{4}=1$ let $P$ be a point in the second quadrant such that the tangent at $P$ to the ellipse is perpendicular to the line …
  2. If an ellipse passes through the point $(4,1)$ and has foci at $(\pm3,0)$, then its eccentricity is
  3. An angle of intersection of the curves, $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ and $x^{2}+y^{2}=ab$, $a>b$, is :
  4. If a tangent to the ellipse $x^{2}+4y^{2}=4$ meets the tangents at the extremities of its major axis at $B$ and $C$, then the circle with $BC$ as diameter passes through …
  5. Let the ellipse $E:x^{2}+9y^{2}=9$ intersect the positive $x$- and $y$-axes at the points $A$ and $B$ respectively. Let the major axis of $E$ be a diameter of the circle …
  6. Let the maximum area of the triangle that can be inscribed in the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{4}=1$, $a>2$, having one of its vertices at one end of the …
  7. Let $x=9$ be a directrix of an ellipse $E$, whose centre is at the origin and eccentricity is $\frac{1}{3}$. Let $P(\alpha,0)$, $\alpha>0$, be a focus of $E$ and $AB$ be …
  8. In a group of $100$ persons $75$ speak English and $40$ speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only …
  9. If the tangents at the points $(2,y)$ on the ellipse, $\frac{x^{2}}{16}+\frac{y^{2}}{4}=1$ meet the tangent at $(4,y_1)$ on it at the points $A$ and $B$, then the length …
  10. Let the tangent and normal at the point $(3\sqrt{3},1)$ on the ellipse $\frac{x^{2}}{36}+\frac{y^{2}}{4}=1$ meet the $y$-axis at the points $A$ and $B$ respectively. Let …
  11. A line passing through the point $P(\sqrt{5},\sqrt{5})$ intersects the ellipse $\frac{x^{2}}{36}+\frac{y^{2}}{25}=1$ at $A$ and $B$ such that $(PA)\cdot(PB)$ is maximum. …
  12. The line $12x\cos\theta+5y\sin\theta=60$ is tangent to which of the following curves ?
  13. Let a circle of radius $4$ be concentric to the ellipse $15x^{2}+19y^{2}=285$. Then the common tangents are inclined to the minor axis of the ellipse at the angle
  14. If the points of intersections of the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{b^{2}}=1$ and the circle $x^{2}+y^{2}=4b$, $b>4$ lie on the curve $y^{2}=3x^{2}$, then $b$ …
  15. Let $E_1:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, $a>b$. Let $E_2$ be another ellipse such that it touches the end points of major axis of $E_1$ and the foci of $E_2$ …
  16. Let $P\left(\frac{2\sqrt{3}}{\sqrt{7}},\frac{6}{\sqrt{7}}\right)$, $Q$, $R$ and $S$ be four points on the ellipse $9x^{2}+4y^{2}=36$. Let $PQ$ and $RS$ be mutually …
  17. Let for two distinct values of $p$ the lines $y=x+p$ touch the ellipse $E:\frac{x^{2}}{4^{2}}+\frac{y^{2}}{3^{2}}=1$ at the points $A$ and $B$. Let the line $y=x$ …
  18. Let the length of a latus rectum of an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ be $10$. If its eccentricity is the minimum value of the function …
  19. Let $A(\alpha,0)$ and $B(0,\beta)$ be the points on the line $5x+7y=50$. Let the point $P$ divide the line segment $AB$ internally in the ratio $7:3$. Let $3x-25=0$ be a …
  20. Let $S_1:x^{2}+y^{2}=9$ and $S_2:(x-2)^{2}+y^{2}=1$. Then the locus of center of a variable circle $S$ which touches $S_1$ internally and $S_2$ externally always passes …
  21. If the radius of the largest circle with centre $(2,0)$ inscribed in the ellipse $x^{2}+4y^{2}=36$ is $r$, then $12r^{2}$ is equal to
  22. If the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ meets the line $\frac x7+\frac{y}{2\sqrt6}=1$ on the $x$-axis and the line $\frac x7-\frac{y}{2\sqrt6}=1$ on the …
  23. Let the eccentricity of an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, $a>b$, be $\frac{1}{4}$. If this ellipse passes through the point …
  24. Let $P(3\cos\alpha,2\sin\alpha)$, $\alpha\neq0$, be a point on the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$, $Q$ be a point on the circle $x^{2}+y^{2}-14x-14y+82=0$ …
  25. The length of the chord of the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{16}=1$, whose mid point is $\left(1,\frac{2}{5}\right)$, is equal to :
  26. Consider ellipses $E_k:kx^{2}+k^{2}y^{2}=1$, $k=1,2,\ldots,20$. Let $C_k$ be the circle which touches the four chords joining the end points (one on minor axis and …
  27. The line $y=x+1$ meets the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{2}=1$ at two points $P$ and $Q$. If $r$ is the radius of the circle with $PQ$ as diameter then …
  28. For which of the following curves, the line $x+\sqrt{3}y=2\sqrt{3}$ is the tangent at the point $\left(\frac{3\sqrt{3}}{2},\frac{1}{2}\right)$ ?
  29. If $S$ and $S'$ are the foci of the ellipse $\frac{x^{2}}{18}+\frac{y^{2}}{9}=1$ and $P$ be a point on the ellipse, then $\min(SP\cdot S'P)+\max(SP\cdot S'P)$ is equal …
  30. Let a tangent be drawn to the ellipse $\frac{x^{2}}{27}+y^{2}=1$ at $(3\sqrt{3}\cos\theta,\sin\theta)$ where $\theta\in\left(0,\frac{\pi}{2}\right)$. Then the value of …
  31. The eccentricity of an ellipse $E$ with centre at the origin $O$ is $\frac{\sqrt{3}}{2}$ and its directrices are $x=\pm\frac{4\sqrt{6}}{3}$. Let …
  32. Let $C$ be the circle of minimum area enclosing the ellipse $E:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ with eccentricity $\frac{1}{2}$ and foci $(\pm2,0)$. Let $PQR$ …
  33. If the curves, $\frac{x^{2}}{a}+\frac{y^{2}}{b}=1$ and $\frac{x^{2}}{c}+\frac{y^{2}}{d}=1$ intersect each other at an angle of $90^{\circ}$, then which of the following …
  34. Let an ellipse $E:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, $a^{2}>b^{2}$, passes through $\left(\sqrt{\frac{3}{2}},1\right)$ and has eccentricity …
  35. Let $f(x)=x^{2}+9$, $g(x)=\frac{x}{x-9}$ and $a=f\circ g(10)$, $b=g\circ f(3)$. If $e$ and $l$ denote the eccentricity and the length of the latus rectum of the ellipse …
  36. Let $P$ be a parabola with vertex $(2,3)$ and directrix $2x+y=6$. Let an ellipse $E:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, $a>b$, of eccentricity …
  37. The locus of mid-points of the line segments joining $(-3,-5)$ and the points on the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{9}=1$ is :
  38. Let $\frac{x^{2}}{f(a^{2}+7a+3)}+\frac{y^{2}}{f(3a+15)}=1$ represent an ellipse with major axis along $y$-axis, where $f$ is a strictly decreasing positive function on …
  39. If the length of the minor axis of an ellipse is equal to half of the distance between the foci, then the eccentricity of the ellipse is :
  40. If the length of the minor axis of an ellipse is equal to one fourth of the distance between the foci, then the eccentricity of the ellipse is :
  41. If $m$ is the slope of a common tangent to the curves $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ and $x^{2}+y^{2}=12$, then $12m^{2}$ is equal to :
  42. Let the eccentricity of the ellipse $x^{2}+a^{2}y^{2}=25a^{2}$ be $b$ times the eccentricity of the hyperbola $x^{2}-a^{2}y^{2}=5$, where $a$ is the minimum distance …
  43. Let $L$ be a tangent line to the parabola $y^{2}=4x-20$ at $(6,2)$. If $L$ is also a tangent to the ellipse $\frac{x^{2}}{2}+\frac{y^{2}}{b}=1$, then the value of $b$ is …
  44. The length of the latus-rectum of the ellipse, whose foci are $(2,5)$ and $(2,-3)$ and eccentricity is $\frac{4}{5}$, is
  45. Let $P$ be a point on the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$. Let the line passing through $P$ and parallel to $y$-axis meet the circle $x^{2}+y^{2}=9$ at point …
  46. If the curve $x^{2}+2y^{2}=2$ intersects the line $x+y=1$ at two points $P$ and $Q$, then the angle subtended by the line segment $PQ$ at the origin is :
  47. The locus of the mid point of the line segment joining the point $(4,3)$ and the points on the ellipse $x^{2}+2y^{2}=4$ is an ellipse with eccentricity :
  48. If the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{b^{2}}=1$ and the circle $x^{2}+y^{2}=4b$ intersect at an angle $\frac{\pi}{3}$, then a value of '$b$' is
  49. The centre of a circle $C$ is at the centre of the ellipse $E:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, $a>b$. Let $C$ pass through the foci $F_1$ and $F_2$ of $E$ …
  50. Let the ellipse $3x^{2}+py^{2}=4$ pass through the centre $C$ of the circle $x^{2}+y^{2}-2x-4y-11=0$ of radius $r$. Let $f_1,f_2$ be the focal distances of the point $C$ …

Other chapters