An ellipse has its center at $(1, -2)$, one focus at $(3, -2)$, and one vertex at $(5, -2)$. Then the length of its latus rectum is :

The coordinates of the foci of the ellipse $16x^{2} + 9y^{2} = 144$ are

The ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and the straight line $y = mx + c$ intersect in real points only if

If an ellipse with foci at $(3,3)$ and $(-4,4)$ is passing through the origin,then the eccentricity of that ellipse is

If the normal at the point $P(\theta)$ to the ellipse $\frac{x^2}{14} + \frac{y^2}{5} = 1$ intersects it again at the point $Q(2\theta)$,then $\cos \theta$ is equal to

Let $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(b < a)$ be an ellipse with major axis $AB$ and minor axis $CD$. Let $F_1$ and $F_2$ be its two foci,with $A, F_1, F_2, B$ in that order on the segment $AB$. Suppose $\angle F_1CB = 90^{\circ}$. The eccentricity of the ellipse is: