Parabola
Tangent Intersection — Coordinate Ratio
nta_pyq_2023_apr
Grade 11
Question:
The ordinates of $P$ and $Q$ on the parabola with focus $(3,0)$ and directrix $x=-3$ are in ratio $3:1$. If $R(\alpha,\beta)$ is the intersection of tangents at $P$ and $Q$, then $\dfrac{\beta^2}{\alpha}$ is equal to
Step-by-Step Solution
Key Concept: Parabola: $y^2=12x$. Parametric points $(3t^2,6t)$. Tangent intersection: $\alpha=3t_1t_2$, $\beta=3(t_1+t_2)$. Given $\frac{6t_1}{6t_2}=3\Rightarrow t_1=3t_2$.
$\frac{\beta^2}{\alpha}=16$.
Correct Answer: 16