A line y = m x + 1 intersects the circle x - 3 2 + y + 2 2 = 25 at the points P and Q . If the midpoint of…

Equation of line $PQ:y=mx+1$ Let $R$ be the mid point of $PQ$, so its $x$-coordinate of midpoint $=-\frac{3}{5}$ Hence, coordinates of midpoint $R\left(-\frac{3}{5},-\frac{3\mathrm{m}}{5}+1\right)$ Let centre of circle is $O\left(3,2\right)$ Since, perpendicular dropped from centre upon any chord divides it in two equal parts. Hence, line $PQ$ will be perpendicular to $RO$ $\Rightarrow m_{RQ}\cdot m_{RO}=-1$ $\Rightarrow m\left(\frac{-\frac{3\mathrm{m}}{5}+1+2}{-\frac{3}{5}-3}\right)=-1$ $\Rightarrow m\left(\frac{-3\mathrm{m}+15}{-18}\right)=-1$ $\Rightarrow m^2-5\mathrm{m}+6=0$ $\Rightarrow m=2, 3$

$\Rightarrow 2\le m<4$