Question 1119858
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According to the description, we are to view the ellipse as having a horizontal major axis.  The standard form of the equation for such an ellipse is<br>
{{{(x-h)^2/a^2+(y-k)^2/b^2 = 1}}}<br>
In that form, the center of the ellipse as (h,k); a and b are the semi-major and semi-minor axes, and c is the distance from the center to each focus, where c^2 = a^2-b^2.<br>
The minimum distance between earth and the moon is 3633 units to the right of earth; the maximum distance between earth and the moon is 4055 units to the left of earth.  That means the length of the major axis is 3633+4055 = 7688 units; then the semi-major axis, a, is 7688/2 = 3844 units.<br>
c, the distance from the center of the ellipse to a focus (the earth), is then 3844-3633 = 211 units.<br>
Then, since the problem specifies that the earth be at the origin, the center of the ellipse is (-211,0).<br>
We have h=-211 and k=0; and we know a=3844, so we can calculate a^2 = 14776336.  Now we need to find b^2 using c^2 = a^2-b^2.<br>
{{{211^2 = 3844^2-b^2}}}
{{{b^2 = 3844^2-211^2 = 14731815}}}<br>
Now we have all the numbers we need to write the equation:<br>
{{{(x-(-211))^2/14776336+(y-0)^2/14731815 = 1}}}<br>
or<br>
{{{(x+211)^2/14776336+y^2/14731815 = 1}}}