Parabola
Focal Chord — Division Point and Perpendicular Line
nta_pyq_2023_apr
Grade 11
Question:
Let $PQ$ be a focal chord of $y^2=36x$ of length 100, making acute angle with $x$-axis. Ordinate of $P$ is positive and $M$ is on $PQ$ with $PM:MQ=3:1$. Which point does NOT lie on the line through $M$ perpendicular to $PQ$?
$(-6,45)$
$(6,29)$
$(3,33)$
$(-3,43)$
Step-by-Step Solution
Key Concept: For $y^2=36x$ ($a=9$): focal chord length $=a(t+\frac{1}{t})^2=100$. $t+\frac{1}{t}=\pm\frac{10}{3}$. For $t=3$: $P=(81,54)$, $Q=(1,6)$.
$M=(21,9)$, perp line: $4x+3y=111$. $(-3,43)$ gives $117\neq111$.
Correct Answer: 4