Let $S$ and $S\,'$ be the foci of an ellipse and $B$ be any one of the extremities of its minor axis. If $\Delta S\,'BS$ is a right angled triangle with right angle at $B$ and area $(\Delta S\,'BS) = 8\,sq.$ units, then the length of a latus rectum of the ellipse is

If the eccentricity of the two ellipse $\frac{{{x^2}}}{{169}} + \frac{{{y^2}}}{{25}} = 1$ and $\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1$ are equal, then the value of $a/b$ 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

Two sets $A$ and $B$ are as under:

$A = \{ \left( {a,b} \right) \in R \times R:\left| {a – 5} \right| < 1 \,\,and\,\,\left| {b – 5} \right| < 1\} $; $B = \left\{ {\left( {a,b} \right) \in R \times R:4{{\left( {a – 6} \right)}^2} + 9{{\left( {b – 5} \right)}^2} \le 36} \right\}$ then : . . . . .

Let $P$ be a variable point on the ellipse $x^2 + 3y^2 = 3$ , then the maximum perpendicular distance of $P$ from the line $x -y = 10$ is