The slopes of three sides of a triangle $ABC$ are $-1, -2, 3$ respectively. If the orthocenter of triangle $ABC$ is origin, then the locus of its centroid is $y = \frac{a}{b}x$ where $a, b$ are relatively prime then $b - a$ is equal to ____.
Step-by-Step Solution
Key Concept: The orthocenter is the intersection of altitudes, each perpendicular to a side, and can be found by solving simultaneous equations of perpendicular lines.
With slopes of $BC$, $CA$, and $AB$ as $-1, -2, -3$ respectively, the orthocenter $H$ is found using perpendicularity conditions. The slopes of altitudes $AH$, $BH$, $CH$ are $1$, $ rac{1}{2}$, $- rac{1}{3}$ respectively. Using the centroid formula and the constraint equations $3h = x_3 + rac{5x_1}{9} + rac{4x_3}{9} = 2x_3$ and $3k = rac{5x_1}{9} + rac{2x_3}{9} - rac{x_3}{9} = rac{4x_3}{9}$, we get $ rac{k}{h} = rac{2}{9}$, yielding $y = rac{2}{9}x$ and $a - b = 9 - 2 = 7$.
Correct Answer: 7