Problem E

Sliding Parabola

Suppose the parabola with equation \(y = 4 - x^2\) has vertex at \(P\) and crosses the \(x\)-axis at points \(A\) and \(B\), with \(B\) lying to the right of \(A\) on the \(x\)-axis.

This parabola is translated so that its vertex moves along the line \(y = x+ 4\) to the point \(Q\). The new parabola crosses the \(x\)-axis at points \(B\) and \(C\), with \(C\) lying to the right of \(B\) on the \(x\)-axis.

Determine the coordinates of \(C\).