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 101-150 of 187

  1. Let $O(0,0)$ and $A(0,1)$ be two fixed points. Then, the locus of a point $P$ such that the perimeter of $\Delta AOP$ is $4$ is
  2. If the curves $\frac{x^{2}}{\alpha}+\frac{y^{2}}{4}=1$ and $y^{3}=16x$ intersect at right angles, then a value of $\alpha$ is :
  3. If $a$ and $c$ are positive real numbers and the ellipse $\frac{x^{2}}{4c^{2}}+\frac{y^{2}}{c^{2}}=1$ has four distinct points in common with the circle …
  4. If tangents are drawn to the ellipse $x^{2}+2y^{2}=2$ at all points on the ellipse other than its four vertices then the mid points of the tangents intercepted between …
  5. A stair-case of length $l$ rests against a vertical wall and a floor of a room. Let $P$ be a point on the stair-case, nearer to its end on the wall, that divides its …
  6. In the ellipse, the distance between its foci is $6$ and minor axis is $8$. Then its eccentricity is :
  7. The eccentricity of an ellipse having centre at the origin, axes along the co-ordinate axes and passing through the points $(4,-1)$ and $(-2,2)$ is
  8. The locus of the foot of perpendicular drawn from the centre of the ellipse $x^{2}+3y^{2}=6$ on any tangent to it is :
  9. Let $\theta$ be the acute angle between the tangents to the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{1}=1$ and the circle $x^{2}+y^{2}=3$ at their point of intersection in …
  10. If the distance between the foci of an ellipse is $6$ and the distance between its directrices is $12$, then the length of its latus rectum is
  11. Equation of the line passing through the points of intersection of the parabola $x^{2}=8y$ and the ellipse $\frac{x^{2}}{3}+y^{2}=1$ is :
  12. The eccentricity of an ellipse whose centre is at the origin is $\frac12$. If one of its directrices is $x=-4$, then the equation of the normal to it at …
  13. Let $S=\left\{(x,y)\in\mathbb R^2:\frac{y^2}{1+r}-\frac{x^2}{1-r}=1\right\}$, where $r\neq\pm1$. Then $S$ represents :
  14. If the tangents on the ellipse $4x^{2}+y^{2}=8$ at the points $(1,2)$ and $(a,b)$ are perpendicular to each other, then $a^{2}$ is equal to
  15. A focus of an ellipse is at the origin. The directrix is the line $x=4$ and the eccentricity is $\frac12$. Then the length of the semi-major axis is :
  16. The equation of the circle passing through the foci of the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$, and having centre at $(0,3)$ is :
  17. If $3x+4y=12\sqrt2$ is a tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{9}=1$ for some $a\in\mathbb R$, then the distance between the foci of the ellipse is :
  18. The foci of the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{b^{2}}=1$ and the hyperbola $\frac{x^{2}}{144}-\frac{y^{2}}{81}=\frac{1}{25}$ coincide. Then the value of $b^{2}$ …
  19. The minimum area of a triangle formed by any tangent to the ellipse $\frac{x^{2}}{16}+\frac{y^{2}}{81}=1$ and the co-ordinate axes is :
  20. An ellipse has $OB$ as semi minor axis, $F$ and $F'$ its focii and the angle $FBF'$ is a right angle. Then the eccentricity of the ellipse is :
  21. If $OB$ is the semi-minor axis of an ellipse, $F_1$ and $F_2$ are its foci and the angle between $F_1B$ and $F_2B$ is a right angle, then the square of the eccentricity …
  22. The eccentricity of an ellipse, with its centre at the origin, is $\frac12$. If one of the directrices is $x=4$, then the equation of the ellipse is :
  23. The ellipse $x^{2}+4y^{2}=4$ is inscribed in a rectangle aligned with the coordinate axes, which in turn is inscribed in another ellipse that passes through the point …
  24. Let the length of the latus rectum of an ellipse with its major axis along $x$-axis and centre at the origin, be $8$. If the distance between the foci of this ellipse is …
  25. In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is $10$ and one of the foci is at $(0,5\sqrt3)$, then the length …
  26. Let the equations of two ellipses be $E_1:\frac{x^{2}}{3}+\frac{y^{2}}{2}=1$ and $E_2:\frac{x^{2}}{16}+\frac{y^{2}}{b^{2}}=1$. If the product of their eccentricities is …
  27. Equation of the ellipse whose axes are the axes of coordinates and which passes through the point $(-3,1)$ and has eccentricity $\sqrt{\frac25}$ is :
  28. The locus of a variable point whose distance from $(-2,0)$ is $\dfrac23$ times its distance from the line $x=-\dfrac92$ is
  29. Let $E$ be the ellipse $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}=1$ and $C$ be the circle $x^{2}+y^{2}=9$. Let $P$ and $Q$ be the points $(1,2)$ and $(2,1)$ respectively. Then
  30. The equation $2x^{2}+3y^{2}-8x-18y+35=k$ represents
  31. Let $P$ be a variable point on the ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$ with foci $F_1$ and $F_2$. If $A$ is the area of the triangle $PF_1F_2$, then …
  32. Find the point on the curve $4x^{2}+a^{2}y^{2}=4a^{2}$, $4<a^{2}<8$ that is farthest from the point $(0,-2)$.
  33. The radius of the circle passing through the foci of the ellipse $\dfrac{x^{2}}{16}+\dfrac{y^{2}}{9}=1$, and having its centre at $(0,3)$ is
  34. If $P=(x,y)$, $F_1=(3,0)$, $F_2=(-3,0)$ and $16x^{2}+25y^{2}=400$, then $PF_1+PF_2$ equals
  35. The number of values of $c$ such that the straight line $y=4x+c$ touches the curve $\left(x^{2}/4\right)+y^{2}=1$ is
  36. Let $P$ be a point on the ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$, $0<b<a$. Let the line parallel to $y$-axis passing through $P$ meet the circle …
  37. On the ellipse $4x^{2}+9y^{2}=1$, the points at which the tangents are parallel to the line $8x=9y$ are
  38. Consider the family of circles $x^{2}+y^{2}=r^{2}$, $2<r<5$. If in the first quadrant, the common tangent to a circle of this family and the ellipse $4x^{2}+25y^{2}=100$ …
  39. An ellipse has eccentricity $\dfrac12$ and one focus at the point $P\left(\dfrac12,1\right)$. Its one directrix is the common tangent, nearer to the point $P$, to the …
  40. Find the co-ordinates of all the points $P$ on the ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$, for which the area of the triangle $PON$ is maximum, where $O$ …
  41. Find the equation of the common tangent in the $1^{st}$ quadrant to the circle $x^{2}+y^{2}=16$ and the ellipse $\dfrac{x^{2}}{25}+\dfrac{y^{2}}{4}=1$. Also find the …
  42. The minimum area of the triangle formed by the tangent to $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$ and the coordinate axes is
  43. The area of the quadrilateral formed by the tangents at the end points of latus rectum to the ellipse $\dfrac{x^2}{9}+\dfrac{y^2}{5}=1$, is
  44. If tangents are drawn to the ellipse $x^{2}+2y^{2}=2$, then the locus of the mid-point of the intercept made by the tangents between the coordinate axes is
  45. Let the line $y-x=1$ intersect the ellipse $\frac{x^2}{2}+\frac{y^2}{1}=1$ at the points $A$ and $B$. Then the angle made by the line segment $AB$ at the center of the …
  46. Let $(h,k)$ lie on the circle $C:x^2+y^2=4$ and the point $(2h+1,3k+2)$ lie on an ellipse with eccentricity $e$. Then the value of $\frac5{e^2}$ is equal to …
  47. Let a focus of the ellipse $E:\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ be $S(4,0)$ and its eccentricity be $\frac45$. If the point $P(3,\alpha)$ lies on $E$ and $O$ is …
  48. Let an ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, $a<b$, pass through the point $(4,3)$ and have eccentricity $\frac{\sqrt5}{3}$. Then the length of its latus …
  49. Let $E$ be the ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$. For any three distinct points $P, Q$ and $Q'$ on $E$, let $M(P,Q)$ be the mid-point of the line segment joining …
  50. If the line $\alpha x+4y=\sqrt7$, where $\alpha\in\mathbf{R}$, touches the ellipse $3x^2+4y^2=1$ at the point $P$ in the first quadrant, then one of the focal distances …

Other chapters