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

The length of the chord of the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$, whose mid point is $\left(1, \frac{2}{5}\right)$, is equal to:

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse $36 x^{2}+4 y^{2}=144$

An ellipse has $OB$ as semi minor axis, $F$ and $F'$ its foci and the angle $FBF'$ is a right angle. Then the eccentricity of the ellipse is

A man running a racecourse notes that the sum of the distances from the two flag posts from him is always $10 \,m$ and the distance between the flag posts is $8\, m$ Find the equation of the posts traced by the man.

The number of $p$ oints which can be expressed in the form $(p_1/q_ 1 ,  p_2/q_2)$, ($p_i$ and $q_i$ $(i = 1,2)$ are co-primes) and lie on the ellipse $\frac{{{x^2}}}{9} + \frac{{{y^2}}}{4} = 1$ is