The sum of the focal distances of the point $\left(\frac{4}{\sqrt{5}}, \frac{3}{\sqrt{5}}\right)$ on the ellipse $9x^2+4y^2=36$ is

Let a tangent drawn at any point on the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ cut the $X$-axis at $Q$. Let $R$ be the image of $Q$ with respect to $y=x$. If $S$ is a circle with $QR$ as its diameter,then the fixed point through which the circle $S$ passes is

If the straight line $x \cos \alpha + y \sin \alpha = p$ touches the curve $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$,then prove that $a^{2} \cos^{2} \alpha + b^{2} \sin^{2} \alpha = p^{2}$.

If a tangent to the ellipse $x^{2}+4y^{2}=4$ meets the tangents at the extremities of its major axis at $B$ and $C$,then the circle with $BC$ as diameter passes through the point:

Let the eccentricity of the ellipse $2x^2 + ay^2 - 8x - 2ay + (8 - a) = 0$ be $\frac{1}{\sqrt{3}}$. If the major axis of this ellipse is parallel to the $Y$-axis,then the equation of the tangent to this ellipse with slope $1$ 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