NIMCET 2010 — Mathematics PYQ

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Question

NIMCET 2010 — Mathematics PYQ

NIMCET | Mathematics | 2010

The angle between the asymptotes of the hyperbola $27x^2 - 9y^2 = 24$ is:

Choose the correct answer:

Explanation

Solving:

1. Standard Form of Hyperbola:

Divide the equation $27x^2 - 9y^2 = 24$ by $24$ :

\frac{27x^2}{24} - \frac{9y^2}{24} = 1

\frac{9x^2}{8} - \frac{3y^2}{8} = 1

\frac{x^2}{8/9} - \frac{y^2}{8/3} = 1

Here, $a^2 = \frac{8}{9}$ and $b^2 = \frac{8}{3}$ .

2. Calculating $a$ and $b$ :

$a = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3}$

$b = \sqrt{\frac{8}{3}} = \frac{2\sqrt{2}}{\sqrt{3}}$

3. Angle Between Asymptotes:

The formula for the angle ( $2\theta$ ) between the asymptotes is:

\tan \theta = \frac{b}{a}

Substitute the values:

\tan \theta = \frac{\frac{2\sqrt{2}}{\sqrt{3}}}{\frac{2\sqrt{2}}{3}}

\tan \theta = \frac{3}{\sqrt{3}} = \sqrt{3}

$\theta = \tan^{-1}(\sqrt{3}) = 60^\circ$

4. Total Angle:

The total angle between asymptotes $= 2\theta$

Angle $= 2 \times 60^\circ = 120^\circ$

Correct Option:

(b) 120°