The equation of tangent and normal at point $(3, -2)$ of ellipse $4{x^2} + 9{y^2} = 36$ are
$\frac{x}{3} - \frac{y}{2} = 1,\;\frac{x}{2} + \frac{y}{3} = \frac{5}{6}$
$\frac{x}{3} + \frac{y}{2} = 1,\;\frac{x}{2} - \frac{y}{3} = \frac{5}{6}$
$\frac{x}{2} + \frac{y}{3} = 1,\;\frac{x}{3} - \frac{y}{2} = \frac{5}{6}$
None of these
The equation of the ellipse whose latus rectum is $8$ and whose eccentricity is $\frac{1}{{\sqrt 2 }}$, referred to the principal axes of coordinates, is
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
The eccentricity of an ellipse whose centre is at the origin is $\frac{1}{2}$ . If one of its directices is $x = - 4$ then the equation of the normal to it at $\left( {1,\frac{3}{2}} \right)$ is
The line $y=x+1$ meets the ellipse $\frac{x^{2}}{4}+\frac{y^{2}}{2}=1$ at two points $P$ and $Q$. If $r$ is the radius of the circle with $PQ$ as diameter then $(3 r )^{2}$ is equal to
The length of the latus rectum of the ellipse $5{x^2} + 9{y^2} = 45$ is