If $x^{2}+9 y^{2}-4 x+3=0, x, y \in R$, then $x$ and $y$ respectively lie in the intervals:

If lines $3x + 2y = 10$ and $-3x + 2y = 10$ are tangents at the extremities of latus rectum of an ellipse whose centre is origin, then the length of latus rectum of ellipse is 

The equation of the normal to the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ at the point $(a\cos \theta ,\;b\sin \theta )$ is

An ellipse is drawn with major and minor axes of lengths $10 $ and $8$ respectively. Using one focus as centre, a circle is drawn that is tangent to the ellipse, with no part of the circle being outside the ellipse. The radius of the circle is

The normal at a point $P$ on the ellipse $x^2+4 y^2=16$ meets the $x$-axis at $Q$. If $M$ is the mid point of the line segment $P Q$, then the locus of $M$ intersects the latus rectums of the given ellipse at the points

If $m$ is the slope of a common tangent to the curves $\frac{x^{2}}{16}+\frac{y^{2}}{9}=1$ and $x^{2}+y^{2}=12$, then $12\; m ^{2}$ is equal to