Question

If the line $y = mx + 7\sqrt{3}$ is normal to the hyperbola $\frac{x^2}{24} - \frac{y^2}{18} = 1,$ then a value of m is

• A. $\frac{\sqrt{5}}{2}$
• B. $\frac{3}{\sqrt{5}}$
• C. $\frac{2}{\sqrt{5}}$
• D. $\frac{\sqrt{15}}{2}$

Answer 1

Answer: (C)

$\frac{x^2}{24} - \frac{y^2}{18} = 1 \Rightarrow \, a = \sqrt{24}; b = \sqrt{18}$
Parametric normal :
$\sqrt{24}cos \theta. x + \sqrt{18}. y cot \theta = 42$
At x = 0 : $y = \frac{42 |}{\sqrt{18}} tan \theta = 7\sqrt{3}$(from given
equation )
$\Rightarrow \, \, tan \theta = \sqrt{\frac{3}{2}} \, \Rightarrow \, sin \theta = \pm \sqrt{3}{5}$
slope of parametric normal = $\frac{-\sqrt{24}cos \theta}{\sqrt{18}cot \theta} = m$
$\Rightarrow \, \, = \, - \sqrt{\frac{4}{3}} sin \theta = - \frac{2}{\sqrt{5}} \, or \, \frac{2}{\sqrt{5}}$