A man running a race course notes that the sum of the distances from the two flag posts from him - Maths - Conic Sections

Question

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

Ajanta Trivedi · Accepted Answer

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 $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ (3) let the major axis of the ellipse be x-axis and the minor axis be y-axis $b^{2} = a^{2} (1 - e^{2})$ where e is the eccentricity of the ellipse from (1): PS+PS'= 2a = length of the major axis 2a=10â‡’a=10/2=5 (4) from (2): SS'=2ae=8 ae=8/2=4 5e=4â‡’e=4/5 $b^{2} = 5^{2} (1 - {(\frac{4}{5})}^{2}) = 25 (1 - \frac{16}{25}) = 25 \times \frac{9}{25} = 9$ thus the equation of the path is $\frac{x^{2}}{5^{2}} + \frac{y^{2}}{3^{2}} = 1 \Rightarrow \frac{x^{2}}{25} + \frac{y^{2}}{9} = 1$ which is the required equation of the path hope this helps you cheers!!

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.