Analytic Geometry – Problem 32: find the area of

Let $H$ be the hyperbola $x^2-\dfrac{y^2}{24}=1$ with foci $F_1(-5,0)$ and $F_2(5,0)$. Point $P$ lies on the right branch of $H$.

If $|PF_1|=\dfrac{4}{3}|PF_2|$, find the area of $\triangle F_1PF_2$.

(1) The hyperbola is $\dfrac{x^2}{1}-\dfrac{y^2}{24}=1$, so $a^2=1\Rightarrow a=1$. Hence for any $P$ on the right branch, $$|PF_1|-|PF_2|=2a=2.$$ (2) Let $|PF_2|=t\,(t>0)$. Then $|PF_1|=\dfrac43 t$, so $$|PF_1|-|PF_2|=\left(\frac43-1\right)t=\frac13 t=2\Rightarrow t=6.$$ Thus $|PF_2|=6$ and $|PF_1|=8$.

(3) The distance between foci is $|F_1F_2|=10$. So the side lengths of $\triangle F_1PF_2$ are $6,8,10$, hence it is a right triangle. Therefore $$S_{\triangle F_1PF_2}=\frac12\cdot 6\cdot 8=24.$$

The area of $\triangle F_1PF_2$ is $24$.