The equation of the normal at the point $(2, 3)$ on the ellipse $9{x^2} + 16{y^2} = 180$, is

If a tangent to the ellipse $x^{2}+4 y^{2}=4$ meets the tangents at the extremities of its major axis at $\mathrm{B}$ and $\mathrm{C}$, then the circle with $\mathrm{BC}$ as diameter passes through the point:

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

For $0 < \theta < \frac{\pi}{2}$, four tangents are drawn at the four points $(\pm 3 \cos \theta, \pm 2 \sin \theta)$ to the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$. If $A(\theta)$ denotes the area of the quadrilateral formed by these four tangents, the minimum value of $A(\theta)$ is

Find the equation for the ellipse that satisfies the given conditions: Vertices $(0,\,\pm 13),$ foci $(0,\,±5)$.

Number of tangents to the circle $x^2 + y^2 = 3$ , which are normal to the ellipse $4x^2 + 9y^2 = 36$ , is