Question

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

• A. $\frac{x^2}{9}+\frac{y^2}{25}=1$
• B. $\frac{x^2}{25}+\frac{y^2}{9}=1$
• C. $\frac{x^2}{3}+\frac{y^2}{5}=1$
• D. None of these

Answer 1

The correct option is B $\frac{x^2}{25}+\frac{y^2}{9}=1$

Let the two flag poles be placed at $S$ and $S^{{}^{\prime }}$

According to the question,

$SP+S^{{}^{\prime }}P=constant=10$

$\therefore$ Path is an ellipse with $a=5$

Also, given $S{S}^{{}^{\prime }}=8$

$\Rightarrow 2ae=8$

$\Rightarrow e=\frac{8}{10}=0.8$

$e=\sqrt{1-\frac{b^2}{a^2}}$

$\Rightarrow 0.8=\sqrt{1-\frac{b^2}{25}}$

$\Rightarrow 0.64==1-\frac{b^2}{25}$

$\Rightarrow \frac{b^2}{25}=0.36$

$\Rightarrow b^2=9$

$\therefore$ Equation of course$=\frac{x^2}{25}+\frac{y^2}{9}=1$