Let $P$ be a variable point on the ellipse $x^2 + 3y^2 = 3$. Then the maximum perpendicular distance of $P$ from the line $x - y = 10$ is (in $\sqrt{2}$)

Let $x = 9$ be a directrix of an ellipse $E$, whose centre is at the origin and eccentricity is $1/3$. Let $P(\alpha, 0), \alpha > 0$, be a focus of $E$ and $AB$ be a chord passing through $P$. Then the locus of the mid point of $AB$ is :

If the chords of contact of tangents drawn from two points $(x_1, y_1)$ and $(x_2, y_2)$ to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ are at right angles,then $\frac{x_1 x_2}{y_1 y_2} = \dots$

An ellipse having foci at $(3, 1)$ and $(1, 1)$ passes through the point $(1, 3)$. Then its eccentricity is

For the ellipse $25x^2 + 9y^2 - 150x - 90y + 225 = 0$,the eccentricity $e$ is: (in $/5$)

Let $C$ be the circle of minimum area enclosing the ellipse $E: \frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with eccentricity $e = \frac{1}{2}$ and foci $(\pm 2, 0)$. Let $PQR$ be a variable triangle,whose vertex $P$ is on the circle $C$ and the side $QR$ of length $2$ is parallel to the major axis of $E$ and contains the point of intersection of $E$ with the negative $y$-axis. Then the maximum area of the triangle $PQR$ is: