Question

A man running around a race course notes that the sum of the distances of two flag posts from him is always $8$ meters. The area of the path he encloses in square meters if the distance between flag posts is 4 meters is

• A. $15\sqrt{3}\pi$ sq.units
• B. $12\sqrt{3}\pi$ sq.units
• C. $18\sqrt{3}\pi$ sq.units
• D. $8\sqrt{3}\pi$ sq.units

Answer 1

The correct option is D $8\sqrt{3}\pi$ sq.units

Locus of a point whose sum of distances from two fixed points(foci) is constant is an ellipse.
Let location of flags is $S$ and $S^{\prime }$
and location of man at any time is $P$
So, $SP+S^{\prime }P=2a$
$\Rightarrow 2a=8\Rightarrow a=4$
distance between flags(foci) is $4$
$\Rightarrow 2ae=4\Rightarrow e=\frac{1}{2},[\because a=4]$
$\because e^2=1-\frac{b^2}{a^2}$
$\Rightarrow \frac{b^2}{a^2}=1-e^2$
$\Rightarrow \frac{b^2}{16}=1-\frac{1}{4}$
$\Rightarrow b^2=12\Rightarrow b=\sqrt{12}$
Since, area of an ellipse $=\pi ab$ sq.units
Area $=\pi \times 4\sqrt{12}$
$=8\sqrt{3}\pi$ sq.units