Ellipse PYQs - Last 5 Years

Consider the ellipses given by
\(x^2+4 y^2=1 \quad \text { and } \quad 4 x^2+y^2=1\)
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 ...

Let $P\left(x_1, y_1\right)$ and $Q\left(x_2, y_2\right)$ 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 $x^2+y^2=9$, and $M$ be the point $(3,0)$.
Suppose...

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 from $S$ to the ellipse, of which one meets the ellipse at one end point ...

Let $T_1$ and $T_2$ be two distinct common tangents to the ellipse $E: \frac{x^2}{6}+\frac{y^2}{3}=1$ and the parabola $P: y^2=12 x$. Suppose that the tangent $T_1$ touches $P$ and $E$ at the points $A_1$ and $A_2$, respectively and the tan...

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 the ellipse and its auxiliary circle at points $E$ and $F$ respe...