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Separated Satellites

2 weather-monitoring satellites orbit the earth.

One is in a circular orbit C₁, the other orbits in an extreme ellipse, C₂ so that it can get closer to the surface.

These orbits can be modelled by the equations below:

\[C_{1}:\left( x + 2\sqrt{17} \right)^{2} + y^{2} = 66\]

\[C_{2}:x = 10\cos t,\ y = 4\sqrt{2}\sin t,\ 0 \leq t \leq 2\pi\]

Give the x coordinate of the points of intersection of the curves C₁ and C₂, given that

\(- 5 \leq x \leq 0\). Give an exact answer in the form

\[A\sqrt{1122} + B\sqrt{17}\]

[7 marks]

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