Let $P$ be a point that varies on the parabola $y^2 = 2x.$ The tangents from $P$ to the circle $(x - 1)^2 + y^2 = 1$ intersect the $y$-axis at $B$ and $C,$ as shown below. If $P = (2,2)$, then find the area of triangle $BCP.$

Question

Let $P$ be a point that varies on the parabola $y^2 = 2x.$  The tangents from $P$ to the circle $(x - 1)^2 + y^2 = 1$ intersect the $y$-axis at $B$ and $C,$ as shown below.  If $P = (2,2)$, then find the area of triangle $BCP.$