# How to solve the word problem below?

Can anyone guide me through this problem? I know how to solve the equation of the circle (the Earth) below but I don't know how to solve the equation of the orbit.

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You can write the formula for the circumference of the Earth as $$(x-a)^2+(y-b)^2=r^2,$$ which is the formula for the circle of radius $r$ centered at the point $(a,b)$. Assuming that the orbit of the satellite is not at an angle with respect to the map, all you need to do is increase the radius.
The radius of the circle is $64$. We may as well move its center to $(0,0)$. The orbit of the satellite then appears as an ellipse with mayor semiaxis $a:=64.6$ and minor semiaxis $b\leq a$. We are not told the direction of these axes. Therefore we shall assume the major axis in direction ${\bf u}:=(\cos\alpha,\sin\alpha)$ and the minor axis in direction ${\bf v}:=(-\sin\alpha,\cos\alpha)$. A parametric representation of the elllipse is then given by $${\bf z}(t):=a \cos t\>{\bf u}+ b\sin t\>{\bf v}\qquad(0\leq t\leq2\pi)\ ,$$ or in coordinates: $$x(t)=a\cos\alpha\cos t-b\sin\alpha\sin t,\quad y(t)=a\sin\alpha\cos t +b\cos\alpha\sin t\ .$$