Straight Lines
Straight Line
star_batch_jee_advanced_2025
Grade None
Question:
A piece of cheese is located at $(12, 10)$ in a coordinate plane. A mouse is at $(4, -2)$ and is running up the line $y = -5x+18$. At the point $(a, b)$, the mouse starts getting farther from the cheese rather than closer to it. The value of $(a+b)$ is:
Step-by-Step Solution
Key Concept: The foot of perpendicular satisfies both the original line equation and the perpendicularity condition using direction ratios.
To find the foot of the perpendicular from $P(12, 10)$ to the line $y + 5x = 18$, we use the condition that $PN \perp$ to the given line. The foot $(a, b)$ satisfies $\frac{a - 12}{5} = \frac{b - 10}{-1}$ (perpendicularity) and lies on the line. Solving these simultaneously gives $(a, b) = (2, 8)$.
Correct Answer: 2