Let the foci of a hyperbola be $(1, 14)$ and $(1, -12)$. If it passes through the point $(1, 6)$, then the length of its latus-rectum is
Step-by-Step Solution
Key Concept: The hyperbola has a vertical transverse axis; identify $c$ from the foci, then use the definition $|PF_1|-|PF_2|=2a$ with $P=(1,6)$ to find $a$, then $b^2=c^2-a^2$, and finally $\ell(LR)=2b^2/a$.
Centre $(1,1)$, $c=13$ (distance from centre to each focus). For $P=(1,6)$: $|PF_1|=8$, $|PF_2|=18$, $2a=|18-8|=10$, $a=5$. $b^2=c^2-a^2=169-25=144$. $\ell(LR)=\dfrac{2b^2}{a}=\dfrac{2\times144}{5}=\dfrac{288}{5}$.
Correct Answer: 4