The midpoint is \((1,9/2)\), and the segment gradient is \((6-3)/(-2-4)=-1/2\). The perpendicular bisector has gradient \(2\), so \[y-\frac92=2(x-1)\quad\Longrightarrow\quad y=2x+\frac52.\] Substitute into the other line: \[x-4\left(2x+\frac52\right)-4=0\iff-7x-14=0.\] Thus \(x=-2\), \(y=-3/2\), giving \(\boxed{P=(-2,-3/2)}\).