Point Q is symmetric to P (4, -1) with respect to the bisector of the first quadrant. If the length of $PQ \text{ is }k\sqrt{2},$ then$k =$ 

It is given that the point Q is symmetric to P ( 4, -1) with respect to $y= x.$ So,coordinates of Q are(-1, 4). 

$\therefore PQ=\sqrt{(4-(-1){)}^2+(-1-4{)}^2}=5\sqrt{2}$