If any tangent to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ cuts off intercepts of length $h$ and $k$ on the axes, then $\frac{{{a^2}}}{{{h^2}}} + \frac{{{b^2}}}{{{k^2}}} = $

Product of slopes of common tangents to the ellipse $\frac{x^2}{32} + \frac{y^2}{8} = 1$ and parabola $y^2 = 8x$ is -

If tangents are drawn to the ellipse $x^2 + 2y^2 = 2$ at all points on the ellipse other than its four vertices than the mid points of the tangents intercepted between the coordinate axes lie on the curve

In a group of $100$ persons $75$ speak English and $40$ speak Hindi. Each person speaks at least one of the two languages. If the number of persons, who speak only English is $\alpha$ and the number of persons who speak only Hindi is $\beta$, then the eccentricity of the ellipse $25\left(\beta^2 x^2+\alpha^2 y^2\right)=\alpha^2 \beta^2$ is $.......$

Tangent is drawn to ellipse $\frac{{{x^2}}}{{27}} + {y^2} = 1$ at $(3\sqrt 3 \cos \theta ,\;\sin \theta )$ where $\theta \in (0,\;\pi /2)$. Then the value of $\theta $ such that sum of intercepts on axes made by this tangent is minimum, is

Let $P(2,2)$ be a point on an ellipse whose foci are $(5,2)$ and $(2,6)$, then eccentricity of ellipse is