If the eccentricity of an ellipse be $1/\sqrt 2 $, then its latus rectum is equal to its

How many real tangents can be drawn to the ellipse $5x^2 + 9y^2 = 32$ from the point $(2,3)$

Let $L$ be a tangent line to the parabola $y^{2}=4 x-20$ at $(6,2)$ . If $L$ is also a tangent to the ellipse $\frac{ x ^{2}}{2}+\frac{ y ^{2}}{ b }=1,$ then the value of $b$ is equal to ..... .

Let $\theta$ be the acute angle between the tangents to the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{1}=1$ and the circle $x^{2}+y^{2}=3$ at their point of intersection in the first quadrant. Then $\tan \theta$ is equal to :

The lengths of major and minor axis of an ellipse are $10$ and $8$ respectively and its major axis along the $y$ - axis. The equation of the ellipse referred to its centre as origin is

The locus of mid points of parts in between axes and tangents of ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ will be