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.
from (2): SS'=2ae=8
thus the equation of the path is
which is the required equation of the path.
hope this helps you.