If $S$ and $S^{\prime}$ are the foci of the ellipse $\frac{x^2}{18}+\frac{y^2}{9}=1$ and $P$ is a point on the ellipse,then $\min \left(SP \cdot S^{\prime}P\right) + \max \left(SP \cdot S^{\prime}P\right)$ is equal to:

Let the line $2x + 3y - k = 0, k > 0$,intersect the $x$-axis and $y$-axis at the points $A$ and $B$,respectively. If the equation of the circle having the line segment $AB$ as a diameter is $x^2 + y^2 - 3x - 2y = 0$ and the length of the latus rectum of the ellipse $x^2 + 9y^2 = k^2$ is $\frac{m}{n}$,where $m$ and $n$ are coprime,then $2m + n$ is equal to

$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(b > a)$ is an ellipse with eccentricity $e = \frac{1}{\sqrt{2}}$. If the angle of intersection between the ellipse and the parabola $y^2 = 4ax$ is $\theta$,then the coordinates of the point corresponding to $\theta$ on the ellipse are:

$A$ point moves such that the sum of its distances from two fixed points $(ae, 0)$ and $(-ae, 0)$ is always $2a$. Then the equation of its locus is

Let $x^2+y^2=20$ be the director circle of an ellipse $E$ whose major axis is the $X$-axis and minor axis is the $Y$-axis. If the length of the latus rectum of $E$ is $2$,then the distance between its foci is

The distance between the foci of an ellipse is $16$ and its eccentricity is $\frac{1}{2}$. The length of the major axis of the ellipse is: