If $OB$ is the semi-minor axis of an ellipse, $F_1$ and $F_2$ are its foci and the angle between $F_1B$ and $F_2B$ is a right angle, then the square of the eccentricity of the ellipse is

The eccentricity of an ellipse is $2/3$, latus rectum is $5$ and centre is $(0, 0)$. The equation of the ellipse is

If $PQ$ is a double ordinate of hyperbola $\frac{{{x^2}}}{{{a^2}}} - \frac{{{y^2}}}{{{b^2}}} = 1$ such that $OPQ$ is an equilateral triangle, $O$ being the centre of the hyperbola. Then the eccentricity $e$ of the hyperbola satisfies

Let an ellipse $E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, a^{2}>b^{2}$, passes through $\left(\sqrt{\frac{3}{2}}, 1\right)$ and has ecentricity $\frac{1}{\sqrt{3}} .$ If a circle, centered at focus $\mathrm{F}(\alpha, 0), \alpha>0$, of $\mathrm{E}$ and radius $\frac{2}{\sqrt{3}}$, intersects $\mathrm{E}$ at two points $\mathrm{P}$ and $\mathrm{Q}$, then $\mathrm{PQ}^{2}$ is equal to:

The equation of the tangent at the point $(1/4, 1/4)$ of the ellipse $\frac{{{x^2}}}{4} + \frac{{{y^2}}}{{12}} = 1$ is

Find the equation for the ellipse that satisfies the given conditions: Length of major axis $26$ foci $(±5,\,0)$