12 A diameter of a circle \(C _ { 1 }\) has end-points at \(( - 3 , - 5 )\) and \(( 7,3 )\).
- Find an equation of the circle \(C _ { 1 }\).
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The circle \(C _ { 1 }\) is translated by \(\binom { 8 } { 4 }\) to give circle \(C _ { 2 }\), as shown in the diagram. - Find an equation of the circle \(C _ { 2 }\).
The two circles intersect at points \(R\) and \(S\). - Show that the equation of the line \(R S\) is \(y = - 2 x + 13\).
- Hence show that the \(x\)-coordinates of \(R\) and \(S\) satisfy the equation \(5 x ^ { 2 } - 60 x + 159 = 0\).
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