Question

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

Answer 1

The equation of posts traced by the man is same as an ellipse equation.

Let the equation be, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

Given,

$2a=10\Rightarrow a=5$

$2c=8\Rightarrow c=4$

formula: $c^2=a^2-b^2$

$\therefore b=\sqrt{a^2-c^2}=\sqrt{25-16}\Rightarrow b=3$

Therefore the required equation is,

$\frac{x^2}{5^2}+\frac{y^2}{3^2}=1$

$\Rightarrow \frac{x^2}{25}+\frac{y^2}{9}=1$