Question

A man running round a racecourse notes that the sum of the distances of two flag-posts from him is always $10$ metres and the distance between the flag-posts is $8$ metres. If the area of the path he encloses in square metres is $k\pi$. Find the value of $k.$

• A. $12$
• B. $13$
• C. $14$
• D. $15$

Answer 1

Answer: (D)

$15$

Clearly, the racecourse will be an ellipse with the flag-posts as its foci.

If $a$ and $b$ are the semi-major and minor axes of the ellipse, then sum of focal distances $=2a=10$ and $2ae=8.$
Also $b=a\sqrt{\left[(1-e^2)\right]}.$ Solving these equations, we get $a=5,e=\frac{8}{2a}=\frac{8}{10}=\frac{4}{5}$ and $b=5\sqrt{(1-\frac{16}{25})}=3$

$\therefore$ Area of the ellipse = $\pi ab=\pi .5.3=15\pi .$ Hence $k=15.$