Chapter 10.4, Problem 46ES

As illustrated in the accompanying figure, suppose that two observers are stationed at the points $F_1\left(c,0\right)\text{and}{F}_2\left(-c,0\right)$ in an $xy\text{-coordinate}$ system. Suppose also that the sound of an explosion in the $xy\text{-plane}$ is heard by the $F_1$ observer t seconds before it is heard by the $F_2$ observer. Assuming that the speed of sound is a constant $v,$ show that the explosion occurred somewhere on the hyperbola

$\frac{x^2}{v^2{t}^2/4}-\frac{y^2}{c^2-\left(v^2{t}^2/4\right)}=1$