Question

A man running in a racecourse notes that the sum of the distances of the two flag posts from him is always 10 m, and the distance between the flag posts is 8 m. Find the equation of the path traced by the man.

Answer 1

We know that an ellipse is the locus of a point that moves in such a way that the sum of its distances from two fixed points (called foci) is constant.

So, the path traced by the man is an ellipse.

Let the equation of the ellipse be

$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,$ where $b^2=a^2(1-e^2).$

Clearly, $2a=10$ and $2ae=8$

$\Rightarrow a=5$ and $e=\frac{4}{5}$

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

$\Rightarrow b=3.$

Hence, the equation of the path is $\frac{x^2}{25}+\frac{y^2}{9}=1.$