A right angled triangle $ABC$ ($\angle C = \frac{\pi}{2}$) is constructed so that its sides are parallel to coordinate axes and the medians through $A$ and $B$ lie on the lines $y = 3x + 1$ and $y = mx + 2$ respectively. Then product of values of $m$ for which such a triangle is possible is ____.
Step-by-Step Solution
Key Concept: The slopes of medians in a right triangle depend on the ratio of the legs; use median formulas connecting vertices to opposite side midpoints.
For a right triangle with legs $a$ and $b$, the medians have slopes $m_1 = \frac{a/2}{b} = 3$ and $m_2 = \frac{a}{b/2}$. From the first equation, $a = 6b$. Computing the ratio: $\frac{m_1}{m_2} = \frac{a/2b}{2a/b} = \frac{a^2}{4ab} = \frac{a}{4b} = \frac{6b}{4b} = \frac{3}{2}$, giving $m_2 = 2m_1/3 = 2$. Alternatively, with $m_1 = \frac{b}{a/2} = 3$ and $m_2 = \frac{b/2}{a}$, we get $\frac{m_1}{m_2} = \frac{2b^2}{ab} = 4$, so $m_2 = 3/4$.
Correct Answer: 9