Question

A man running round a race course notes that the sum of the distances of two flag-posts from him is always 10 meters and the distance between the flag posts is 8 metres. The area of the path he encloses in square metres is

• A. $15\pi$
• B. $12\pi$
• C. $18\pi$
• D. $8\pi$

Answer 1

Answer: (A)

$15\pi$

Suppose the race course is an ellipse with flag-posts as its foci.
If a and b are the semi-major and minor axes of the ellipse, then 2a = 10 and 2ae = 8
$\Rightarrow a=5,e=\frac{4}{5}.$
Then, $b^2=a^2(1-e^2)$
$\Rightarrow b^2=25(1-\frac{16}{25})=9\Rightarrow b=3.$
Therefore the area of the ellipse $=\pi ab=15\pi$.