Answer
See steps of explanations.
Work Step by Step
Step 1. Start with the initial equation $\sqrt {(x+c)^2+(y)^2}-\sqrt {(x-c)^2+(y)^2}=\pm2a$, rewrite the equation as $\sqrt {(x+c)^2+(y)^2}=\pm2a+\sqrt {(x-c)^2+(y)^2}$
Step 2. Take the square for both sides to get $(x+c)^2+(y)^2=4a^2+(x-c)^2+(y)^2\pm4a\sqrt {(x-c)^2+(y)^2}$
Step 3. Isolate the radical and simplify other terms to get $xc-a^2=\pm a\sqrt {(x-c)^2+(y)^2}$
Step 4. Take the square for both sides to get $x^2c^2+a^4-2xca^2=a^2((x-c)^2+(y)^2)$
Step 5. Expand the right side and combine like terms for $x^2$ and $y^2$ to get
$x^2c^2+a^4-2xca^2=a^2x^2-2xca^2+a^2c^2+a^2y^2$ and
$(c^2-a^2)x^2-a^2y^2=a^2(c^2-a^2)$ which is the same as the equation in the Exercise.
