In coordinate space, let A be the intersection point of the line $\frac { x } { 2 } = y = z + 3$ and the plane $\alpha : x + 2 y + 2 z = 6$. A sphere with center at point $( 1 , - 1,5 )$ passing through point A intersects plane $\alpha$ to form a figure with area $k \pi$. Find the value of $k$. [3 points]