The position of the point $(1, 3)$ with respect to the ellipse $4x^2 + 9y^2 - 16x - 54y + 61 = 0$ is:

The foci of the ellipse $25x^2 + 4y^2 + 100x - 4y + 100 = 0$ are

In an ellipse with its centre at the origin,if the difference between the lengths of the major axis and the minor axis is $10$ and one of the foci is at $(0, 5\sqrt{3})$,then the length of its latus rectum is:

Let an ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a^{2}>b^{2}$,pass through $\left(\sqrt{\frac{3}{2}}, 1\right)$ and have eccentricity $e = \frac{1}{\sqrt{3}}$. If a circle,centered at the focus $F(\alpha, 0), \alpha > 0$ of $E$ and having radius $r = \frac{2}{\sqrt{3}}$,intersects $E$ at two points $P$ and $Q$,then $PQ^{2}$ is equal to:

If $F_1$ and $F_2$ are the foci of the ellipse $16 x^2+25 y^2=400$ and $P$ is any point on it,then the value of the product $P F_1 \cdot P F_2$ lies in the interval

If $F_1$ and $F_2$ are the feet of the perpendiculars from the foci $S_1$ and $S_2$ of an ellipse $\frac{x^2}{5} + \frac{y^2}{3} = 1$ on the tangent at any point $P$ on the ellipse,then $(S_1 F_1) (S_2 F_2)$ is equal to