Question

A man running a race-course note that the sum of the distance from two flag posts from him is always 10 meters and the distance between the flag posts is 8 meters. The equation of the path traced by the man is given by

• A. $\frac{x^2}{9}+\frac{y^2}{25}=1$
• B. $\frac{x^2}{9}+\frac{y^2}{16}=1$
• C. $\frac{x^2}{25}+\frac{y^2}{9}=1$
• D. $\frac{x^2}{16}+\frac{y^2}{25}=1$

Answer 1

Answer: (C)

Clearly, path traced by the man will be ellipse.
Given, $SP+S^{\prime }P=10$
i.e., $2a=10\Rightarrow a=5$
Since, the coordinates of $S$ and $S^{\prime }$ are $(c,0)$ and $(-c,0)$ Therefore, distance between $S$ and $S^{\prime }$ is $2c=8$

$\Rightarrow c=4$
$\because c^2=a^2-b^2$
$\therefore 16=25-b^2$
$\Rightarrow b^2=25-16=9$
$\Rightarrow b=3$
Hence, equation of path (ellipse) is
$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\Rightarrow \frac{x^2}{25}+\frac{y^2}{9}=1$