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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.

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The radius of orbit is 0.6 units more than radius of earth and centre is same as that of earth

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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: $$x(t)=a\cos\alpha\cos t-b\sin\alpha\sin t,\quad y(t)=a\sin\alpha\cos t +b\cos\alpha\sin t\ .$$

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