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, $SP+ S'P = 10$
i.e., $2a = 10 \: \Rightarrow \:\: a = 5$
Since, the coordinates of $S$ and $S' $ are $(c, 0)$ and $(-c, 0)$ Therefore, distance between $S$ and $S'$ 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 $