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

let the flag posts be at point S and S'.

since the sum of the distances from two fixed points is constant, therefore the travelled path is elliptical.

here S and S' are the foci of the ellipse.

such that PS+PS'=10 m...........(1)

the distance between the flag post is SS'= 8 m........(2)

i.e. the distance between two foci is 8 m.

let the equation of the ellipse is ..........(3)

let the major axis of the ellipse be x-axis and the minor axis be y-axis.

where e is the eccentricity of the ellipse.

from (1): PS+PS'= 2a = length of the major axis.

2a=10⇒a=10/2=5.........(4)

from (2): SS'=2ae=8

ae=8/2=4

5e=4⇒e=4/5

thus the equation of the path is

which is the required equation of the path.

hope this helps you.

cheers!!

**
**