Parabola
Conic from Slope Condition — Focal Distance
nta_pyq_2024_apr
Grade 11
Question:
Let a conic $C$ pass through the point $(4,-2)$ and $P(x,y)$, $x\geq3$, be any point on $C$. Let the slope of the line touching $C$ only at a single point $P$ be half the slope of the line joining the points $P$ and $(3,-5)$. If the focal distance of the point $(7,1)$ on $C$ is $d$, then $12d$ equals
Step-by-Step Solution
Key Concept: The condition $\frac{dy}{dx}=\frac{1}{2}\cdot\frac{y+5}{x-3}$ gives the ODE $\frac{2dy}{y+5}=\frac{dx}{x-3}$. Integrate: $2\ln|y+5|=\ln|x-3|+C$. Using point $(4,-2)$: $C=2\ln3$.
Parabola $(y+5)^2=9(x-3)$, $a=9/4$. Focal distance of $(7,1)$: $d=25/4$. $12d=75$.
Correct Answer: 75