The length of the latus rectum of the ellipse $9x^2 + 4y^2 = 1$ is

$A$ 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

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 $A, A^{\prime}$ be the end points of the major axis,$S, S^{\prime}$ be the foci,and $B, B^{\prime}$ be the end points of the minor axis of an ellipse $E$. If $\angle BAB^{\prime}=60^{\circ}$,then find $\angle SBS^{\prime}$.

$A$ point $(4, -1)$ lies on an ellipse whose axes are parallel to the coordinate axes. If the line $x + 4y - 10 = 0$ is tangent to the ellipse at this point,find its equation $(a > b)$.

If the line $2x - 3y + 4 = 0$ cuts the ellipse $x = 3 \cos \theta, y = 5 \sin \theta$ at points $A$ and $B$,and $(\alpha, \beta)$ is the midpoint of $\overline{AB}$,then $3\beta - 2\alpha =$