Question

If \(\theta\) is the acute angle between the asymptotes of a hyperbola \(7x^2 - 9y^2 = 63\), then \(\cos \theta =\)

• A. \(\dfrac{1}{4}\)
• B. \(\dfrac{3}{4}\)
• C. \(\dfrac{1}{8}\)
• D. \(\dfrac{1}{2}\)

Hint:

Use the formula \(\cos \theta = \dfrac{a^2 - b^2}{a^2 + b^2}\) for hyperbola asymptote angle.

Answer 1

The Correct Option is C

For hyperbola of the form \(\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1\), angle between asymptotes: \(\cos \theta = \dfrac{a^2 - b^2}{a^2 + b^2}\).
Given: \(7x^2 - 9y^2 = 63 \Rightarrow \dfrac{x^2}{9} - \dfrac{y^2}{7} = 1\), so \(a^2 = 9, b^2 = 7\).
\[ \cos \theta = \dfrac{9 - 7}{9 + 7} = \dfrac{2}{16} = \dfrac{1}{8} \]