The sum of focal distances of any point on the ellipse with major and minor axes as $2a$ and $2b$ respectively, is equal to

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 line $12 x \,\cos \theta+5 y \,\sin \theta=60$ is tangent to which of the following curves?

Find the equation for the ellipse that satisfies the given conditions: Major axis on the $x-$ axis and passes through the points $(4,\,3)$ and $(6,\,2)$

The centre of an ellipse is $C$ and $PN$ is any ordinate and $A$, $A’$ are the end points of major axis, then the value of $\frac{{P{N^2}}}{{AN\;.\;A'N}}$ is

If the chord through the point whose eccentric angles are $\theta \,\& \,\phi $ on the ellipse,$(x^2/a^2) + (y^2/b^2) = 1$  passes through the focus, then the value of $ (1 + e)$ $\tan(\theta /2) \tan(\phi /2)$ is