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

An ellipse passes through the foci of the hyperbola $9x^2 - 4y^2 = 36$,and its major and minor axes lie along the transverse and conjugate axes of the hyperbola,respectively. If the product of the eccentricities of the two conics is $\frac{1}{2}$,then which of the following points does not lie on the ellipse?

Let $f(x)=x^2+9$,$g(x)=\frac{x}{x-9}$,$a=f(g(10))$,and $b=g(f(3))$. If $e$ and $l$ denote the eccentricity and the length of the latus rectum of the ellipse $\frac{x^2}{a}+\frac{y^2}{b}=1$,then $8e^2+l^2$ is equal to.

The eccentricity of the ellipse $9x^{2} + 25y^{2} = 225$ is

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 the $y$-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 $PQ$ such that $PR:RQ=4:3$ as $P$ moves on the ellipse,is:

The eccentricity of the conic $36x^2 + 144y^2 - 36x - 96y - 119 = 0$ is