Let $P$ be an arbitrary point on the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ where $a > b > 0$. Suppose $F_1$ and $F_2$ are the foci of the ellipse. The locus of the centroid of the $\triangle P F_1 F_2$ as $P$ moves on the ellipse is

Lines $x+y=1$ and $3y=x+3$ intersect the ellipse $x^{2}+9y^{2}=9$ at the points $P, Q$ and $R$. The area of the $\triangle PQR$ is

The area enclosed by the ellipse $4x^2 + 9y^2 = 1$ is . . . . . . sq. units.

The number of tangents to the circle $x^2 + y^2 = 3$ that are normal to the ellipse $4x^2 + 9y^2 = 36$ is

Let the equation of an ellipse be $\frac{x^{2}}{144}+\frac{y^{2}}{25}=1$. Then,the radius of the circle with center $(0, \sqrt{2})$ and passing through the foci of the ellipse is

Let the ellipse $E : x^2 + 9y^2 = 9$ intersect the positive $x$- and $y$-axes at the points $A$ and $B$ respectively. Let the major axis of $E$ be a diameter of the circle $C$. Let the line passing through $A$ and $B$ meet the circle $C$ at the point $P$. If the area of the triangle with vertices $A, P$ and the origin $O$ is $\frac{m}{n}$,where $m$ and $n$ are coprime,then $m - n$ is equal to