The equation $2x^2 + 3y^2 - 8x - 18y + 35 = k$ represents:

If lines $3x + 2y = 10$ and $-3x + 2y = 10$ are tangents at the extremities of the latus rectum of an ellipse whose centre is the origin,then the length of the latus rectum of the ellipse is:

The line passing through the extremity $A$ of the major axis and extremity $B$ of the minor axis of the ellipse $x^2+9y^2=9$ meets its auxiliary circle at the point $M$. Then the area of the triangle with vertices at $A$,$M$,and the origin $O$ 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:

Point $O$ is the centre of the ellipse with major axis $AB$ and minor axis $CD$. Point $F$ is one focus of the ellipse. If $OF = 6$ and the diameter of the inscribed circle of triangle $OCF$ is $2$,then the product $(AB)(CD)$ is equal to

If the line $x - 2y = 12$ is tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at the point $(3, -4.5)$,then the length of the latus rectum of the ellipse is: