For the ellipse $\frac{{{x^2}}}{{64}} + \frac{{{y^2}}}{{28}} = 1$, the eccentricity is

A tangent is drawn to the ellipse $\frac{{{x^2}}}{{32}} + \frac{{{y^2}}}{8} = 1$ from the point $A(8, 0)$ to touch the ellipse at point $P.$ If the normal at $P$ meets the major axis of ellipse at point $B,$ then the length $BC$ is equal to (where $C$ is centre of ellipse) - ............ $\mathrm{units}$

The eccentricity of the ellipse $\frac{{{{(x - 1)}^2}}}{9} + \frac{{{{(y + 1)}^2}}}{{25}} = 1$ is

How many real tangents can be drawn to the ellipse $5x^2 + 9y^2 = 32$ from the point $(2,3)$

Let $P$ be a point on the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$. Let the line passing through $P$ and parallel to $y$-axis meet the circle $x^2+y^2=9$ at point $Q$ such that $P$ and $Q$ are on the same side of the $x$-axis. Then, the eccentricity of the locus of the point $R$ on $P Q$ such that $P R: R Q=4: 3$ as $P$ moves on the ellipse, is :

An arch is in the form of a semi-cllipse. 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.