The product of the lengths of perpendiculars from the foci on any tangent to the ellipse $3x^2 + 5y^2 = 1$ is:

The position of the point $(4, -3)$ with respect to the ellipse $2x^2 + 5y^2 = 20$ is

The equation of the ellipse in the standard form whose length of the latus rectum is $4$ and whose distance between the foci is $4 \sqrt{2}$,is

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 $/3$)

Given the ellipse $(E) 4x^2 + 9y^2 - 36 = 0$,the circle $(C) x^2 + y^2 - 9 = 0$ and two points $A(1, 2)$,$B(2, 1)$,which of the following is correct?

Let $A=\{(\alpha, \beta) \in R \times R :|\alpha-1| \leq 4 \text{ and }|\beta-5| \leq 6\}$ and $B=\left\{(\alpha, \beta) \in R \times R : 16(\alpha-2)^2+9(\beta-6)^2 \leq 144\right\}$. Then