Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
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 than $b - a$ is equal to ________.
Step-by-Step Solution
Key Concept: Slope relationships between sides determine point coordinates; the centroid divides the triangle into specific ratios.
With slopes of $BC$, $CA$, $AB$ as $-1, -2, 3$ respectively, find coordinates of $A$, $B$, $C$ using the slope relationships. From $9y_2 = 4y_3$ and $9y_1 = 5y_3$, combined with the centroid formula $3h = x_1 + \frac{5x_3}{9} + \frac{4x_3}{9} = 2x_3$, we get $\frac{k}{h} = \frac{2}{9}$, so $y = \frac{2}{9}x$ and $b - a = 7$.
Correct Answer: 7