Hyperbola
Point on Hyperbola — Area of Triangle with Foci
nta_pyq_2024_jan
Grade 11
Question:
Let $P$ be a point on the hyperbola $H:\dfrac{x^2}{9}-\dfrac{y^2}{4}=1$, in the first quadrant such that the area of triangle formed by $P$ and the two foci of $H$ is $2\sqrt{13}$. Then the square of the distance of $P$ from the origin is
Step-by-Step Solution
Key Concept: Foci at $(\pm ae,0)=(\pm\sqrt{13},0)$. $|S_1S_2|=2\sqrt{13}$. Area$=\frac{1}{2}\cdot|S_1S_2|\cdot|y_P|=2\sqrt{13}\Rightarrow|y_P|=2$. Substitute $y=2$ in hyperbola to find $x$. Compute $x^2+y^2$.
$e=\sqrt{13}/3$. Foci: $(\pm\sqrt{13},0)$. $|y_P|=2$. Hyperbola: $\frac{x^2}{9}-1=1\Rightarrow x^2=18$. $|OP|^2=18+4=22$.
Correct Answer: 3