Exercise 63 Page 821

At its closest point, Earth is $91.8$ million miles from the center of the Sun, and at its farthest point $94.9$ million miles from the center of the Sun. We want to find the equation for the orbit of Earth, assuming that its center is the origin and that the Sun lies on $x\text{-}$axis. Let's take a look at the described situation.

The orbit of Earth is an ellipse. To find its equation we need to understand that Sun is not the center of this ellipse. Recall the general equation for ellipses centered at the origin. In our case the Sun is situated on the $x\text{-}$ axis, so the ellipse would be horizontal. In such equations, $2a$ is the length of the major axis and $2b$ is the length of the minor axis. We can find $a$ by taking half the length of the major axis, which is the sum of the minimum and maximum distance between the Earth and Sun. Let's also calculate $a^2,$ since it will be necessary in our equation. Recall that the units are millions of miles, so $a=93.35\times 10^6.$ Now, consider that in our case the Sun is one of the foci of the ellipse. From the picture, we can see that we can calculate $c$ by subtracting $a$ from the maximum distance between the Earth and Sun. Now we can use the relationship between $a,$ $b,$ and $c$ to find $b^2.$ Note that $c$ is also measured in million miles. Using $a^2$ and $b^2$ from our calculations above, we can write the equation for the Earth's orbit around the Sun.