Straight Lines
Locus Problems
Premium Question
Grade 11
Question:
The locus of the foot of the perpendicular from the origin to a variable line passing through the fixed point $(a, b)$ is:
(1) $x^2 + y^2 = ax + by$
(2) $x^2 + y^2 + ax + by = 0$
(3) $(x - a)^2 + (y - b)^2 = a^2 + b^2$
(4) $ax + by = 1$
Step-by-Step Solution
Key Concept: Let the foot be $(h, k)$. Since $O, (h, k)$, and $(a, b)$ are such that the segment $O(h, k)$ is perpendicular to $(h, k)(a, b)$: use $\overrightarrow{OH} \cdot \overrightarrow{HF} = 0$.
The detailed step-by-step mathematical proof is available inside the Mathbee app workspace.
Correct Answer: (1)