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
$\frac{{{x^2}}}{{18}} + \frac{{{y^2}}}{{32}} = 1$
$\frac{{{x^2}}}{8} + \frac{{{y^2}}}{9} = 1$
$\frac{{{x^2}}}{{64}} + \frac{{{y^2}}}{{32}} = 1$
$\frac{{{x^2}}}{{16}} + \frac{{{y^2}}}{{24}} = 1$
An ellipse having foci at $(3, 3) $ and $(- 4, 4)$ and passing through the origin has eccentricity equal to
The equation of the ellipse whose vertices are $( \pm 5,\;0)$ and foci are $( \pm 4,\;0)$ is
If the angle between the lines joining the end points of minor axis of an ellipse with its foci is $\pi\over2$, then the eccentricity of the ellipse 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
In an ellipse, with centre at the origin, if the difference of the lengths of major axis and minor axis is $10$ and one of the foci is at $(0, 5\sqrt 3 )$, then the length of its latus rectum is