Question

A man running around a racecourse notes that the sum of the distances of two flag-posts from him is always $20$ meters and distance between the flag-posts is $16$ meters. If the area of the path enclosed in square meters is $k\times 10\pi$, find $k$.

Answer 1

The two flag poles are the foci of the ellipse since it is given sum of distances is fixed.
Distance between foci $=2ae=16$
Sum of distances between him and poles$=2a=20$

$a=10$
$e=\frac{16}{2\ast 10}=0.8$ and $e^2=1-\frac{b^2}{a^2}$
Solving for $b$ we get $b=6$
Area covered by him is nothing but the area of ellipse $=ab\pi =10\times 6\times \pi =60\pi$
$k\times 10\pi =60\pi \Rightarrow k=6$