Let $(x, y)$ be a variable point on the curve $4x^2 + 9y^2 - 8x - 36y + 15 = 0$. Then,$\min (x^2 - 2x + y^2 - 4y + 5) + \max (x^2 - 2x + y^2 - 4y + 5)$ is

The lines $y=2x+\sqrt{76}$ and $2y+x=8$ touch the ellipse $\frac{x^2}{16}+\frac{y^2}{12}=1$. If the point of intersection of these two lines lies on a circle whose centre coincides with the centre of that ellipse,then the equation of that circle is

Tangents are drawn from points on the circle $x^2 + y^2 = 49$ to the ellipse $\frac{x^2}{25} + \frac{y^2}{24} = 1$. The angle between the tangents is:

Let $S = 0$ be an ellipse whose vertices are the extremities of the minor axis of the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ where $a > b$. If $S = 0$ passes through the foci of $E$,then its eccentricity is (considering the eccentricity of $E$ as $e$).

The eccentricity of an ellipse passing through $(3 \sqrt{2}, \sqrt{10})$ with foci at $(-4,0)$ and $(4,0)$ is

The minimum length of the intercept between the coordinate axes made by a tangent of the ellipse $\frac{x^2}{64}+\frac{y^2}{4}=1$ is