Problem

The equation of a circle is $(x - a)^2 + (y - 3)^2 = 20$. The line $y = \frac12 x + 6$ is a tangent to the circle at the point $P$.

(a) Show that one possible value of $a$ is $4$ and find the other possible value.

(b) For $a = 4$, find the equation of the normal to the circle at $P$.

(c) For $a = 4$, find the equations of the two tangents to the circle which are parallel to the normal found in (b).