Coordinate Geometry
Hyperbola branches; angle ACB; eccentricity bound
MJMT_Full_Test_07
Grade 12
Question:
Let $H_L$ and $H_R$ be two branches of a hyperbola with $A,B$ as endpoints of latus rectum on $H_L$. If $C$ is on $H_R$ such that $\angle ACB$ is never obtuse, then maximum possible eccentricity is
Step-by-Step Solution
Key Concept: For $\angle ACB$ never obtuse: $\vec{CA}\cdot\vec{CB}\geq0$ for all $C$ on $H_R$. The latus rectum endpoints $A,B$ subtend a right angle at $C$ when $C$ is on a specific circle. Condition: circle must not intersect $H_R$.
Maximum eccentricity $=\sqrt3$... Answer (B)? But key says C. $e_{\max}=2$.
Correct Answer: 3