- A circle \(C\) has equation \(x ^ { 2 } + y ^ { 2 } - 6 x - 14 y = 40\).
The line \(l\) has equation \(y = x + k\), where \(k\) is a constant.
a. Show that the \(x\)-coordinate of the points where \(C\) and \(l\) intersect are given by the solutions to the equation
$$2 x ^ { 2 } + ( 2 k - 20 ) x + k ^ { 2 } - 14 k - 40 = 0$$
b. Hence find the two values of \(k\) for which \(l\) is a tangent to \(C\).