Ellipse — JEE Advanced PYQs

Consider the ellipses given by

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 P , then the value of …

Let $P\left(x_1,y_1\right)$ and $Q\left(x_2,y_2\right)$ be two distinct points on the ellipse

such that $y_1>0$, and $y_2>0$. Let C denote the circle $x^2+y^2=9$, and $M$ be the point ( 3,0 ).
Suppose the line …

Consider the ellipse $\frac{{\mathrm{x}}^2}{9}+\frac{{\mathrm{y}}^2}{4}=1$. Let $\mathrm{S}(\mathrm{p},\mathrm{q})$ be a point in the first quadrant such that $\frac{{\mathrm{p}}^2}{9}+\frac{{\mathrm{q}}^2}{4}>1$. Two tangents are drawn from S to the ellipse, of which one …

Consider the ellipse

Let $H(\alpha ,0),0<\alpha <2$, be a point. A straight line drawn through $H$ parallel to the $y$-axis crosses the ellipse and its auxiliary circle at points $E$ and $F$ respectively, in the first quadrant. The tangent t …

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 P and Q , and $\mathrm{M}\left(\mathrm{P},{\mathrm{Q}}^{\prime }\right)$ be the mid-point of the line segment joining …

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 $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ passes through the point P. If the tangents to the parabola and the ellipse at th …

Define the collections $\left\{{\mathrm{E}}_1,{\mathrm{E}}_2,{\mathrm{E}}_3,\dots ..\right\}$ of ellipses and $\left\{{\mathrm{R}}_1,{\mathrm{R}}_2,{\mathrm{R}}_3,\dots ..\right\}$ of rectangles as follows : …

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 $Q$. Consider the ellipse whose center is at the origin $O(0,0)$ and whose semi-major axis is $OQ$. If …

Let $F_1\left(x_1,0\right)$ and $F_2\left(x_2,0\right)$, for $x_1<0$ and $x_2>0$, be the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{8}=1$. Suppose a parabola having vertex at the origin and focus at $F_2$ intersects the ellipse at point $M$ in …

Let $F_1\left(x_1,0\right)$ and $F_2\left(x_2,0\right)$, for $x_1<0$ and $x_2>0$, be the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{8}=1$. Suppose a parabola having vertex at the origin and focus at $F_2$ intersects the ellipse at point $M$ in …

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 the circle $x^2+(y-1{)}^2=2$. The straight line $x+y=3$ touches the curves …

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 $P$ and $Q$ meet at the point $R$. If $\Delta (h)=$ area of the triangle …

The ellipse $E_1:\frac{x^2}{9}+\frac{y^2}{4}=1$ is inscribed in a rectangle $R$ whose sides are parallel to the coordinate axes. Another ellipse $E_2$ passing through the point (0, 4) circumscribes the rectangle $R$. The eccentricity of the ellipse $E_2$ is

Tangents are drawn from the point P(3, 4) to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ touching the ellipse at points A and B.

The coordinates of A and B are

Tangents are drawn from the point P(3, 4) to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ touching the ellipse at points A and B.

The orthocenter of the triangle PAB is

Tangents are drawn from the point P(3, 4) to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$ touching the ellipse at points A and B.

The equation of the locus of the point whose distances from the point P and the line AB are equal, is

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 latus rectums of the given ellipse at the points

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 along the coordinate axes, then

The line passing through the extremity $A$ of the major axis and extremity $B$ of the minor axis of the ellipse

meets its auxiliary circle at the point $M$. Then the area of the triangle with vertices at $A,M$ and the origin $O$ is