The equation of line $A B$ is
$4 x+7 y+5=0$ ...(i)
and equation of line $P Q$ is
$2 x-y=0$ ... (ii)
On solving Eqs. (i) and (ii), we get
$x=-\frac{5}{18}$ and $y=-\frac{5}{9}$
$\therefore $ Coordinates of $Q=\left(-\frac{5}{18},-\frac{5}{9}\right)$
Length of $Q P =\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}$
$=\sqrt{\left(1+\frac{5}{18}\right)^{2}+\left(2+\frac{5}{9}\right)^{2}}$
$=\sqrt{\left(\frac{23}{18}\right)^{2}+\left(\frac{23}{9}\right)^{2}}$
$=\frac{23}{9} \sqrt{\frac{1}{4}+1}=\frac{23}{9} \times \frac{\sqrt{5}}{2}$
$=\frac{23}{18} \sqrt{5}\, sq\,$ units
