If the curves $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ and $\frac{x^2}{25}+\frac{y^2}{16}=1$ cut each other orthogonally,then $a^2-b^2$ equals to

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 ellipse having its foci $(0, \pm 1)$ and major axis of length $\sqrt{5}$ is

Tangents are drawn to the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ at all the four ends of its latus rectum. The area (in sq units) of the quadrilateral formed by these tangents is

Let $S(1,0)$ and $S^{\prime}(0,1)$ be the foci of an ellipse such that $SP+S^{\prime} P=2$ for any point $P$ on the ellipse. If $A(x_1, y_1)$ and $A^{\prime}(x_2, y_2)$ are the end points of the major axis of this ellipse,then $x_1+x_2=$

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 :