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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 151-187 of 187

  1. Let $S$ and $S'$ be the foci of the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1$ and $P(\alpha,\beta)$ be a point on the ellipse in the first quadrant. If …
  2. If the points of intersection of the ellipses $x^2+2y^2-6x-12y+23=0$ and $4x^2+2y^2-20x-12y+35=0$ lie on a circle of radius $r$ and centre $(a,b)$, then the value of …
  3. Let each of the two ellipses $E_1:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $(a>b)$ and $E_2:\frac{x^2}{A^2}+\frac{y^2}{B^2}=1$, $(A<B)$ have eccentricity $\frac45$. Let the …
  4. Let the length of the latus rectum of an ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, $(a>b)$, be $30$. If its eccentricity is the maximum value of the function …
  5. An ellipse has its center at $(1,-2)$, one focus at $(3,-2)$ and one vertex at $(5,-2)$. Then the length of its latus rectum is :
  6. Let $P(x_1,y_1)$ and $Q(x_2,y_2)$, $y_1<0$, $y_2<0$, be the end points of the latus rectum of the ellipse $x^2+4y^2=4$. The equations of parabolas with latus rectum $PQ$ …
  7. The line passing through the extremity $A$ of the major axis and extremity $B$ of the minor axis of the ellipse $$x^2+9y^2=9$$ meets its auxiliary circle at the point …
  8. The normal at a point $P$ on the ellipse $x^2+4y^2=16$ meets the $x$-axis at $Q$. If $M$ is the mid point of the line segment $PQ$, then the locus of $M$ intersects the …
  9. An ellipse intersects the hyperbola $2x^2-2y^2=1$ orthogonally. The eccentricity of the ellipse is reciprocal of that of the hyperbola. If the axes of the ellipse are …
  10. The ellipse $E_1:\dfrac{x^2}{9}+\dfrac{y^2}{4}=1$ is inscribed in a rectangle $R$ whose sides are parallel to the coordinate axes. Another ellipse $E_2$ passing through …
  11. A vertical line passing through the point $(h,0)$ intersects the ellipse $\frac{x^2}{4}+\frac{y^2}{3}=1$ at the points $P$ and $Q$. Let the tangents to the ellipse at …
  12. Let $E_1$ and $E_2$ be two ellipses whose centers are at the origin. The major axes of $E_1$ and $E_2$ lie along the $x$-axis and the $y$-axis, respectively. Let $S$ be …
  13. Consider two straight lines, each of which is tangent to both the circle $x^2+y^2=\frac{1}{2}$ and the parabola $y^2=4x$. Let these lines intersect at the point $Q$. …
  14. Let $a,b$ and $\lambda$ be positive real numbers. Suppose $P$ is an end point of the latus rectum of the parabola $y^2=4\lambda x$, and suppose the ellipse …
  15. The length of the chord of the ellipse $\frac{x^2}{4}+\frac{y^2}{2}=1$, whose mid-point is $\left(1,\frac12\right)$, is :
  16. Let $E_1:\frac{x^2}{9}+\frac{y^2}{4}=1$ be an ellipse. Ellipses $E_i$'s are constructed such that their centres and eccentricities are same as that of $E_1$, and the …
  17. Let $P$ be the point in the first quadrant where the given ellipses intersect. If $\theta$ is the acute angle between the tangents to the given ellipses at the point …
  18. For $a=\sqrt2$, if a tangent is drawn to a suitable conic (Column 1) at the point of contact $(-1,1)$, then which of the following options is the only CORRECT …
  19. The orthocentre of the triangle $F_1MN$ is
  20. The coordinates of $A$ and $B$ are
  21. If the tangents to the ellipse at $M$ and $N$ meet at $R$ and the normal to the parabola at $M$ meets the $x$-axis at $Q$, then the ratio of area of the triangle $MQR$ …
  22. The orthocenter of the triangle $PAB$ is
  23. The tangent to a suitable conic (Column 1) at $\left(\sqrt3,\frac12\right)$ is found to be $\sqrt3x+2y=4$, then which of the following options is the only CORRECT …
  24. Let $d$ be the perpendicular distance from the centre of the ellipse $\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$ to the tangent drawn at a point $P$ on the ellipse. If …
  25. Let $ABC$ be an equilateral triangle inscribed in the circle $x^{2}+y^{2}=a^{2}$. Suppose perpendiculars from $A,B,C$ to the major axis of the ellipse …
  26. Prove that, in an ellipse, the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact meet on the …
  27. An ellipse is drawn by taking a diameter of the circle $(x-1)^2+y^2=1$ as its semi-minor axis and a diameter of the circle $x^2+(y-2)^2=4$ as its semi-major axis. If the …
  28. A tangent to the ellipse $x^{2}+4y^{2}=4$ meets the ellipse $x^{2}+2y^{2}=6$ at $P$ and $Q$. Prove that the tangents at $P$ and $Q$ of the ellipse $x^{2}+2y^{2}=6$ are …
  29. If the area of the triangle formed by the tangent at a point in the first quadrant on the ellipse, $4x^{2}+3y^{2}=12$ and its axes is $4$ sq. units, then one such point …
  30. Let the line $2x+3y-k=0$, $k>0$, intersect the $x$-axis and $y$-axis at the points $A$ and $B$, respectively. If the equation of the circle having the line segment $AB$ …
  31. Which of the following points lies on the locus of the foot of perpendicular drawn upon any tangent to the ellipse, $\dfrac{x^{2}}{4}+\dfrac{y^{2}}{2}=1$ from any of its …
  32. Let $E:\dfrac{x^{2}}{a^{2}}+\dfrac{y^{2}}{b^{2}}=1$, $a>b$ and $H:\dfrac{x^{2}}{A^{2}}-\dfrac{y^{2}}{B^{2}}=1$. Let the distance between the foci of $E$ and the foci of …
  33. The equation of the chord, of the ellipse $\dfrac{x^{2}}{25}+\dfrac{y^{2}}{16}=1$, whose mid-point is $(3,1)$ is :
  34. If the midpoint of a chord of the ellipse $\dfrac{x^{2}}{9}+\dfrac{y^{2}}{4}=1$ is $\left(\sqrt{2},\dfrac{4}{3}\right)$, and the length of the chord is …
  35. The area (in sq. units) of the quadrilateral formed by the tangents at the end points of the latera recta to the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{5}=1$, is :
  36. A point on the ellipse, $4x^{2}+9y^{2}=36$, where the normal is parallel to the line, $4x-2y-5=0$, is :
  37. Consider an ellipse, whose center is at the origin and its major axis is along the $x$-axis. If its eccentricity is $\frac35$ and the distance between its foci is $6$ …

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