The point $(1,3)$ with respect to the ellipse $4x^2+9y^2-16x-54y+61=0$ lies

The focus of an ellipse is at the origin. The line $x = 4$ is a directrix and the eccentricity is $1/2$. Find the length of the major axis. (in $/3$)

Let $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b)$ be a given ellipse whose latus rectum length is $10$. If its eccentricity $e$ is the maximum value of the function $\phi(t) = \frac{5}{12} + t - t^{2}$,then $a^{2} + b^{2}$ is equal to:

If the distance of a point on the ellipse $\frac{x^2}{6} + \frac{y^2}{2} = 1$ from the center is $2$,find its eccentric angle $\varphi$.

Find the equation of the ellipse whose vertices are $(\pm 13, 0)$ and foci are $(\pm 5, 0)$.

The locus of the feet of the perpendiculars from either focus of the ellipse $(x - y + 1)^2 + (2x + 2y - 6)^2 = 20$ onto any tangent is: