Parabola
Chord with Given Midpoint — Equation of Chord
nta_pyq_2024_apr
Grade None
Question:
Let $PQ$ be a chord of the parabola $y^2=12x$ and the midpoint of $PQ$ be at $(4,1)$. Then, which of the following points lies on the line passing through the points $P$ and $Q$?
$(3,-3)$
$(2,-9)$
$\left(\dfrac{3}{2},-16\right)$
$\left(\dfrac{1}{2},-20\right)$
Step-by-Step Solution
Key Concept: Use $T=S_1$ for chord with midpoint $(x_1,y_1)$: $yy_1-6(x+x_1)=y_1^2-12x_1$. With $(x_1,y_1)=(4,1)$: $y-6(x+4)=1-48\Rightarrow6x-y=23$.
Chord equation $6x-y=23$. Point $(1/2,-20)$ satisfies it.
Correct Answer: 4