The ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ $(b>a)$ and the parabola $y^2=4ax$ intersect at right angles. If $e$ is the eccentricity of the ellipse,then $2e^2=$

Let $e_1$ and $e_2$ be the eccentricities of the ellipse $\frac{x^2}{b^2} + \frac{y^2}{25} = 1$ and the hyperbola $\frac{x^2}{16} - \frac{y^2}{b^2} = 1$,respectively. If $b < 5$ and $e_1 e_2 = 1$,then the eccentricity of the ellipse having its axes along the coordinate axes and passing through all four foci (two of the ellipse and two of the hyperbola) is:

For the hyperbola $H : x^{2} - y^{2} = 1$ and the ellipse $E : \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ where $a > b > 0$,let $(1)$ the eccentricity of $E$ be the reciprocal of the eccentricity of $H$,and $(2)$ the line $y = \sqrt{\frac{5}{2}} x + K$ be a common tangent of $E$ and $H$. Then $4(a^{2} + b^{2})$ is equal to:

How many parabolas can be drawn if the endpoints of the latus rectum are given?

The eccentricity of the conic $\frac{5}{r}=2+3 \cos \theta+4 \sin \theta$ is

Find the length of the common chord of the ellipse $\frac{(x - 2)^2}{9} + \frac{(y + 2)^2}{4} = 1$ and the circle $x^2 + y^2 - 4x + 2y + 4 = 0$.