Hyperbola
Asymptotes
Premium Question
Grade 11
Question:
The angle between the asymptotes of the hyperbola $\frac{x^2}{4} - \frac{y^2}{9} = 1$ is:
(1) $2 \tan^{-1}\left(\frac{3}{2}\right)$
(2) $2 \tan^{-1}\left(\frac{2}{3}\right)$
(3) $\tan^{-1}\left(\frac{12}{5}\right)$
(4) $\pi - 2 \tan^{-1}\left(\frac{3}{2}\right)$
Step-by-Step Solution
Key Concept: Asymptotes: $y = \pm(b/a)x = \pm(3/2)x$. The angle between them: $\tan \theta = \frac{2(b/a)}{1-(b/a)^2} = \frac{3}{1-9/4} = \frac{3}{-5/4} = -12/5$. So the acute angle between asymptotes is $\pi - \arctan(12/5)$; the full angle between the two branches is $2 \tan^{-1}(3/2)$.
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Correct Answer: (1)