The length of the latus rectum of the ellipse $5x^2 + 9y^2 = 45$ 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 $a^{2} \cos^{2} \alpha + b^{2} \sin^{2} \alpha = p^{2}$.

The parametric representation of a point on the ellipse whose foci are $(-1,0)$ and $(7,0)$ and eccentricity $1/2$ is

If the eccentricities of the two ellipses $\frac{x^2}{169} + \frac{y^2}{25} = 1$ and $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ are equal,then the value of $a/b$ is

Find the number of points on the ellipse $\frac{x^{2}}{50} + \frac{y^{2}}{20} = 1$ from which a pair of perpendicular tangents can be drawn to the ellipse $\frac{x^{2}}{16} + \frac{y^{2}}{9} = 1$.

The eccentricity of the conic $4x^2 + 16y^2 - 24x - 3y = 1$ is