NJC Tutorial Curve Sketching Q7

A satellite is orbiting sound around the planet Earth. Taking the plane of its orbit as the \(x\mathrm{-}y\) plane, the position of the satellite is at the variable point \(P\) with coordinates \(\left( a\cos \theta ,\,b\sin \theta \right)\), where \(a\) and \(b\) are positive constants such that \(a>b\).

1. 1. It is known that the Earth is positioned at the fixed point \({{F}_{1}}\) with coordinates \(\left( \sqrt{{{a}^{2}}-{{b}^{2}}},0 \right)\).
      Show that the distance between the Earth and the satellite can be expressed as \(a-\left( \sqrt{{{a}^{2}}-{{b}^{2}}} \right)\cos \theta \).
   2. The mirror image of the point \({{F}_{1}}\) about the origin is denoted by the point \({{F}_{2}}\).
      Show that the sum of the distances \(P{{F}_{1}}\) and \(P{{F}_{2}}\) is independent of \(\theta \).
2. Show that the satellite’s orbit around the Earth is elliptical.