Answer
$\frac{x^2}{1455642} +\frac{y^2}{1451610}=1$
Work Step by Step
Use the figure given in the Exercise.
Step 1. Assume the orbit of Apollo 11.has a standard ellipse equation $\frac{x^2}{a^2} +\frac{y^2}{b^2}=1 $ with a center at the middle point between the Perilune and the Apolune.
Step 2. Identify the given quantities: radius of the moon $r=1075mi$, the center of the moon is at one focus of the ellipse orbit. At Perilune, the distance to the moon center is $d_1=r+68=1143mi$. At Apolune, the distance to the moon center is $d_2=r+195=1270mi$.
Step 3. The total distance $d_1+d_2=2a=2413$, thus $a=1206.5$. And for the moon center $c=a-d_1=63.5mi$
Step 4. Use the relationship $b^2=a^2-c^2$, we have $b^2=(1206.5)^2-(63.5)^2=1451610$
Step 5. Conclusion: the equation of the orbit can be written as $\frac{x^2}{1455642} +\frac{y^2}{1451610}=1$
