If a number of ellipses are described having the same major axis $2a$ but a variable minor axis,then the tangents at the ends of their latera recta pass through fixed points which are:

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

If an ellipse with its axes as coordinate axes,$2a$ and $2b$ as the lengths of its major and minor axes respectively,passes through the points $(2,2)$ and $(3,1)$,then $3a^2+5b^2=$

If $\frac{\sqrt{3}}{a}x + \frac{1}{b}y = 2$ touches the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$,then its eccentric angle $\theta$ is equal to: ................ $^o$

An arch is in the form of a semi-ellipse. It is $8 \, m$ wide and $2 \, m$ high at the centre. Find the height of the arch at a point $1.5 \, m$ from one end. (in $, m$)

Let $A(\alpha, 0)$ and $B(0, \beta)$ be the points on the line $5x + 7y = 50$. Let the point $P$ divide the line segment $AB$ internally in the ratio $7:3$. Let $3x - 25 = 0$ be a directrix of the ellipse $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and the corresponding focus be $S$. If from $S$,the perpendicular on the $x$-axis passes through $P$,then the length of the latus rectum of $E$ is equal to