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 the line segment $P Q$ has x-coordinate $- \frac{3}{5},$ then which one of the following options is correct ?

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$4 \leq m \leq 6$

$- 3 \leq m < - 1$

$2 \leq m < 4$

$6 \leq m < 8$

answer is C.

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Detailed Solution

$P Q ⊥ Q R \Rightarrow S t o p e O R = - \frac{1}{m} = \frac{- \frac{3}{5} m + 1 + 2}{- \frac{3}{5} - 3}$
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$\Rightarrow m^{2} - 5 m + 6 = 0$
$\Rightarrow m = 2, 3$

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