Question

A man running a race course notes that the sum of the distance from 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 posts traced by the man.

Answer 1

Let $A$ and $B$ be the positions of the two flag posts and $P(x,y)$ be the position of the man such that
$PA+PB=10.$

We know that " If a point moves in a plane in such a way that the sum of its distance from two fixed points is constant then the path is an ellipse and this constant value is equal the length of the major axis of the ellipse".

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

Length of the major axis $=10m$

Points $A$ and $B$ are the foci.

Taking the origin of the coordinate plane as the centre of the ellipse while taking the major axis along the $x$-axis the ellipse.
The equation of then ellipse will be of the form $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ where $a$ is the semi-major axis.
$2a=10$

$\Rightarrow a=5$
Distance between the foci $\left(2c\right)=8$
$\Rightarrow c=ae=4$

Since, $c=\sqrt{a^2-b^2}$, we get
$4=\sqrt{25-b^2}$
$\Rightarrow 16=25-b^2$
$\Rightarrow b^2=25-16=9$
$\Rightarrow b=3$
So, the equation of the path traced by the man is $\frac{x^2}{25}+\frac{y^2}{9}=1$