5. Circles \(C _ { 1 } , C _ { 2 }\) and \(C _ { 3 }\) have collinear centres.
\(C _ { 3 }\) is tangent to both \(C _ { 1 }\) and \(C _ { 2 }\).
The equations for \(C _ { 1 }\) and \(C _ { 2 }\) are as follows:
\(C _ { 1 } : x ^ { 2 } + y ^ { 2 } + 4 x - 60 = 0\)
\(C _ { 2 } : x ^ { 2 } + y ^ { 2 } - 14 x + 40 = 0\)
Find all possible equations for \(C _ { 3 }\).
[0pt] [BLANK PAGE] \section*{6. In this question you must show detailed reasoning.} You are given that \(P _ { 0 } , P _ { 1 }\) and \(Q\) are the vertices of a right-angled triangle with hypotenuse \(P _ { 0 } Q\). The length of \(P _ { 0 } Q\) is 1 .
\(P _ { 2 }\) is the foot of the perpendicular from \(P _ { 1 }\) to \(P _ { 0 } Q\).
\(P _ { 3 }\) is the foot of the perpendicular from \(P _ { 2 }\) to \(P _ { 1 } Q\).
The infinite set of points \(P _ { 4 } , P _ { 5 } , P _ { 6 } , \ldots\) is defined similarly.
The angle at \(P _ { 0 }\) is \(\theta\).
The diagram below shows the points up to \(P _ { 5 }\) :
\includegraphics[max width=\textwidth, alt={}, center]{9345ffb5-b2ec-4366-8956-c8d766bacbd4-12_848_1065_662_571}

1. Given that $$\sum _ { r = 0 } ^ { \infty } P _ { 2 r } P _ { 2 r + 1 } = 2$$ find the value of \(\theta\).
2. For this value of \(\theta\), evaluate the following: $$\sum _ { r = 0 } ^ { \infty } P _ { 2 r + 1 } P _ { 2 r + 2 }$$
3. If instead \(\theta = 45 ^ { \circ }\), determine the smallest value of \(n\) for which: $$\sum _ { r = 0 } ^ { n } P _ { 2 r + 1 } P _ { 2 r + 2 } > 0.999$$ [BLANK PAGE]
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