In the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$, the equation of diameter conjugate to the diameter $y = \frac{b}{a}x$, is

A triangle is formed by the tangents at the point $(2,2)$ on the curves $y^2=2 x$ and $x^2+y^2=4 x$, and the line $x+y+2=0$. If $r$ is the radius of its circumcircle, then $r ^2$ is equal to $........$.

If the normal at one end of the latus rectum of an ellipse  $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ passes through one end of the minor axis then :

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

Find the coordinates of the foci, the vertices, the length of major axis, the minor axis, the eccentricity and the length of the latus rectum of the ellipse $\frac{x^{2}}{4}+\frac{y^2} {25}=1$.

$P$ is a variable point on the ellipse $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ with $AA'$ as the major axis. Then the maximum value of the area of $\Delta APA'$ is