Angle subtended by $AB$ at centre of the hyperbola is
Step-by-Step Solution
Key Concept: The angle subtended by a focal chord at the center of a hyperbola is determined by the eccentricity, which relates the semi-major and semi-minor axes through the geometric properties of the hyperbola.
For a hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, points $A$ and $B$ lie on the hyperbola. The angle subtended at the center $O$ is found using the focal chord property and the eccentricity relationship. Using the standard result for a chord $AB$ on a hyperbola where the chord passes through a focus, if the semi-latus rectum and eccentricity give $e = \frac{5}{3}$, then through parametric analysis with $e^2 = 1 + \frac{b^2}{a^2}$, the angle $\theta$ subtended satisfies $\sin\theta = \frac{3}{5}$, making the answer $\sin^{-1}\frac{3}{5}$.
Correct Answer: 3