If from point $P$ (4, 4) perpendiculars to the straight lines $3x + 4y + 5 = 0$ and $y = mx + 7$ meet at $Q$ and $R$ respectively and area of triangle $PQR$ is maximum. Then the value of $12m$ must be ____.
Step-by-Step Solution
Key Concept: In a cyclic quadrilateral, opposite angles sum to $180°$, and perpendicularity of sides gives the slope relationship.
Since $PQRS$ is a cyclic quadrilateral with rectangle properties, the area of triangle $PQR$ is maximized when $\angle QPR = \angle QSR = 90°$. Using the perpendicularity condition $m_{QS} \cdot m_{SR} = -1$, we get $\left(-\frac{3}{4}\right)m = -1$, so $m = \frac{4}{3}$. Therefore $12m = 12 \cdot \frac{4}{3} = 16$.
Correct Answer: 16