$P$ is a variable point on the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ with $AA'$ as the major axis. Then the maximum value of the area of $\Delta APA'$ is

An ellipse has $OB$ as semi-minor axis,$S$ and $S^{\prime}$ are foci and angle $\angle SBS^{\prime}$ is a right angle. Then the eccentricity of the ellipse is

The sum of focal distances of any point on the ellipse with major and minor axes as $2a$ and $2b$ respectively,is equal to

Consider the parabola $P : y^2 = 4x$ and the ellipse $E : \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$. Let the line segment joining the points of intersection of $P$ and $E$ be their common latus rectum. If the eccentricity of $E$ is $e$,then $e^2 + 2\sqrt{2}$ is equal to . . . . . .

The sides of the rectangle of greatest area that can be inscribed in the ellipse $x^2+4y^2=64$ are:

If $\alpha, \beta$ are the two real roots of the $4^{th}$ roots of unity and $\gamma, \delta$ are the other two roots,then the sum of the eccentricities of the conics $|z-\alpha|+|z-\beta|=4$ and $|z-\gamma|+|z-\delta|=6$ is