8 The lines \(L _ { 1 }\) and \(L _ { 2 }\) are parallel.
\(L _ { 1 }\) has equation
$$5 x + 3 y = 15$$
and \(L _ { 2 }\) has equation
$$5 x + 3 y = 83$$
\(L _ { 1 }\) intersects the \(y\)-axis at the point \(P\).
The point \(Q\) is the point on \(L _ { 2 }\) closest to \(P\), as shown in the diagram.
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-10_849_917_945_561}
8
- Find the coordinates of \(Q\).
8
- (ii) Hence show that \(P Q = k \sqrt { 34 }\), where \(k\) is an integer to be found.
8
- A circle, \(C\), has centre ( \(a , - 17\) ).
\(L _ { 1 }\) and \(L _ { 2 }\) are both tangents to \(C\).
8 - Find \(a\).
8
- (ii) Find the equation of \(C\).
\includegraphics[max width=\textwidth, alt={}, center]{22ff390e-1360-43bd-8c7f-3d2b58627e91-13_2493_1732_214_139}