A man running round a race-course notes that the sum of the distance of two flag-posts from him is always $10\ metres$ and the distance between the flag-posts is $8\ metres$. The area of the path he encloses in square metres is

If the area of the auxiliary circle of the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1\left( {a > b} \right)$ is twice the area of the ellipse, then the eccentricity of the ellipse is

Let the line $2 \mathrm{x}+3 \mathrm{y}-\mathrm{k}=0, \mathrm{k}>0$, intersect the $\mathrm{x}$-axis and $\mathrm{y}$-axis at the points $\mathrm{A}$ and $\mathrm{B}$, respectively. If the equation of the circle having the line segment $\mathrm{AB}$ as a diameter is $\mathrm{x}^2+\mathrm{y}^2-3 \mathrm{x}-2 \mathrm{y}=0$ and the length of the latus rectum of the ellipse $\mathrm{x}^2+9 \mathrm{y}^2=\mathrm{k}^2$ is $\frac{\mathrm{m}}{\mathrm{n}}$, where $\mathrm{m}$ and $\mathrm{n}$ are coprime, then $2 \mathrm{~m}+\mathrm{n}$ is equal to

If the normal at an end of a latus rectum of an ellipse passes through an extremity of the minor axis, then the eccentricity $e$ of the ellipse satisfies

If the latus rectum of an ellipse be equal to half of its minor axis, then its eccentricity is

Let $\mathrm{E}$ be an ellipse whose axes are parallel to the co-ordinates axes, having its center at $(3,-4)$, one focus at $(4,-4)$ and one vertex at $(5,-4) .$ If $m x-y=4, m\,>\,0$ is a tangent to the ellipse $\mathrm{E}$, then the value of $5 \mathrm{~m}^{2}$ is equal to $.....$