Question

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

• A. $15\pi$
• B. $20\pi$
• C. $27\pi$
• D. $30\pi$

Answer 1

Answer: (A)

$15\pi$

Let $P(x,y)$ be the position of the man at any time.
Since $SP+S^{\prime }P=10$ i.e., a constant.

The locus of $P$ is an ellipse with $S$ and $S^{\prime }$ as foci
$\Rightarrow ae=4$, and $2a=10$
$\Rightarrow e=\frac{4}{5}$
Now,

$b^2=a^2(1-e^2)$
$\Rightarrow b^2=25(1-\frac{16}{25})=9$
$\Rightarrow b=3$

Hence, the area of the ellipse

$=\pi ab=\pi \times 5\times 3=15\pi$

Hence, option A.