Parabola
Tangent Lines from External Point — Isosceles Triangle
nta_pyq_2024_apr
Grade 11
Question:
Let $L_1,L_2$ be the lines passing through the point $P(0,1)$ and touching the parabola $9x^2+12x+18y-14=0$. Let $Q$ and $R$ be the points on the lines $L_1$ and $L_2$ such that $\triangle PQR$ is an isosceles triangle with base $QR$. If the slopes of the lines $QR$ are $m_1$ and $m_2$, then $16(m_1^2+m_2^2)$ is equal to
Step-by-Step Solution
Key Concept: Rewrite parabola: $9x^2+12x+4=-18(y-1)\Rightarrow(3x+2)^2=-18(y-1)$, i.e. $(x+2/3)^2=-2(y-1)$. Tangent lines from $P(0,1)$: $y=mx+1$ giving $m=0$ or $m=-4/3$.
$m_1=-1/2$, $m_2=2$. $16(m_1^2+m_2^2)=68$.
Correct Answer: 68