Answer
$x^2-y^2=2.3\times10^{19}$
Work Step by Step
Step 1. Assume the comet following a standard hyperbola equation $\frac{x^2}{a^2} -\frac{y^2}{b^2}=1 $
Step 2. Use the figure given in the Exercise, the two asymptotes form a right angle, meaning that $a=b$
Step 3. As the Sun is at a focus, the minimum distance is $c-a=2\times10^9mi=2bil$ mi
Step 4. Use the formula $c^2=a^2+b^2$, we have $(a+2bil)^2=2a^2$ or $a+2bil=\sqrt 2 a$ or
$a=\frac{2\times10^9}{\sqrt 2 -1}mi$ thus $a^2=b^2\approx2.3\times10^{19}$
Step 5. We can write the equation of the comet as $x^2-y^2=2.3\times10^{19}$
