- A circle \(C\) with radius \(r\)
- lies only in the 1st quadrant
- touches the \(x\)-axis and touches the \(y\)-axis
The line \(l\) has equation \(2 x + y = 12\)
- Show that the \(x\) coordinates of the points of intersection of \(l\) with \(C\) satisfy
$$5 x ^ { 2 } + ( 2 r - 48 ) x + \left( r ^ { 2 } - 24 r + 144 \right) = 0$$
Given also that \(l\) is a tangent to \(C\),
- find the two possible values of \(r\), giving your answers as fully simplified surds.