Question

A line $y = mx + 1$ intersects the circle $(x ? 3)^2 + (y + 2)^2 = 25$ at the points $P$ and $Q$ If the midpoint of the line segment $PQ$ has $x$-coordinate $-\frac{3}{5}$, then which one of the following options is correct?

• A. $-3\le m
• B. $2\le m<4$
• C. $4\le m<6$
• D. $6\le m<8$

Answer 1

Answer: (B)

$2\le m<4$

Answer 2

The mid-point of chord PQ can be considered as $A\left(-\frac{3}{5}, -\frac{3}{5}m+1\right)$
$\because AO \bot PQ Slope of AO ? Slope of PQ = -1$
$\Rightarrow\left(\frac{1-\frac{3}{5}m+2}{-\frac{3}{5}-3}\right).m=-1$
$\Rightarrow \left(3-\frac{3}{5}m\right)m=\frac{18}{5}$
$\Rightarrow m^{2}-5m+6=0$
$\Rightarrow m=2$ or $3$