Ellipse PYQs - Last 10 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...

Let E be the ellipse \({{{x^2}} \over {16}} + {{{y^2}} \over 9} = 1\). 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 M(P, Q') be the mid-point of the line segment joini...

Define the collections {E1, E2, E3, ...} of ellipses and {R1, R2, R3.....} of rectangles as follows :\({E_1}:{{{x^2}} \over 9} + {{{y^2}} \over 4} = 1\)R1 : rectangle of largest area, with sides parallel to the axes, inscribed in E1;En : el...

Consider two straight lines, each of which is tangent to both the circle x2 + y2 = (1/2) and the parabola y2 = 4x. Let these lines intersect at the point Q. Consider the ellipse whose centre is at the origin O(0, 0) and whose semi-major axi...

Let S be the circle in the XY-plane defined the equation x2 + y2 = 4.Let P be a point on the circle S with both coordinates being positive. Let the tangent to S at P intersect the coordinate axes at the points M and N. Then, the mid-point o...

For how many values of p, the circle x2 + y2 + 2x + 4y \(-\) p = 0 and the coordinate axes have exactly three common points?