Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade 11
Question:
The lines $x + y = 0$, $-4y = 0$ and $2x - y = 0$ are the altitudes of a triangle. If one of the vertices has coordinates of the form $\lambda, -\lambda)$, if the locus of the centroid of such a triangle is $ax + by = 0$, then the value of $a + b$ is ______.
Step-by-Step Solution
Key Concept: The centroid of a triangle is the average of its three vertices' coordinates.
The equations of lines $AD$, $BE$, $CF$ are $x + y = 0$, $x - 4y = 0$, $2x - y = 0$ respectively. The centroid $G$ satisfies $x = \frac{\lambda + 4\mu + \delta}{3}$ and $y = \frac{\mu - \lambda + 2\delta}{3}$ where $A(\lambda, -\lambda)$, $B(4\mu, \mu)$, $C(\delta, 2\delta)$. Solving the system yields the centroid coordinates.
Correct Answer: 6