A man running a racecourse notes that the sum of the distances from the two flag posts form him is always 10 m and the distance between the flag posts is 8 m.

Solution:

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

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

Therefore, the path described by the man is an ellipse where the length of the major axis is 10 m, while 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 can be diagrammatically represented as

The equation of the ellipse will be of the form $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$, where $a$ is the semi-major axis

Accordingly, $2 a=10 \Rightarrow a=5$

Distance between the foci $(2 c)=8$

$\Rightarrow c=4$

On using the relation $c=\sqrt{a^{2}-b^{2}}$, we obtain

$4=\sqrt{25-b^{2}}$

$\Rightarrow 16=25-b^{2}$

$\Rightarrow b^{2}=25-16=9$

$\Rightarrow b=3$

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