In an ellipse, the distance between its foci is $6$ and minor axis is $8$. Then its eccentricity is
$\frac{4}{5}$
$\frac{1}{2}$
$\frac{3}{5}$
$\frac{1}{{\sqrt 5 }}$
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
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 $y$-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 $P Q$ such that $P R: R Q=4: 3$ as $P$ moves on the ellipse, is :
If $\alpha $ and $\beta $ are the eccentric angles of the extremities of a focal chord of an ellipse, then the eccentricity of the ellipse is
The distance between the focii of the ellipse $(3x - 9)^2 + 9y^2 =(\sqrt 2 x + y +1)^2$ is-
A circle has the same centre as an ellipse and passes through the foci $F_1 \& F_2$ of the ellipse, such that the two curves intersect in $4$ points. Let $'P'$ be any one of their point of intersection. If the major axis of the ellipse is $17 $ and the area of the triangle $PF_1F_2$ is $30$, then the distance between the foci is :