Question

Equations of tangents to the hyperbola $4x^2-3y^2=24$ which makes an angle ${30}^o$ with $y-$axis are

• A. $\sqrt{3}x-y=\pm \sqrt{10}$
• B. $\sqrt{3}x-y=\pm 10$
• C. $\sqrt{3}x-y=\pm 5$
• D. $\sqrt{3}x-y=\pm \sqrt{5}$

Answer 1

The correct option is A $\sqrt{3}x-y=\pm \sqrt{10}$

Given that:

$4x^2-3y^2=24$

On dividing by $24$, we get

$\frac{x^2}{{\left(\sqrt{6}\right)}^2}-\frac{y^2}{{\left(2\sqrt{2}\right)}^2}=1$

$a=\sqrt{6},b=2\sqrt{2}$

Now,

Equation of tangent

$y=mx\pm \sqrt{a^2{m}^2-b^2}$

As the line makes an angle of ${30}^{\circ }$ from $y-$axis,

$\therefore m=\tan {120}^{\circ }=-\cot {30}^{\circ }=-\sqrt{3}$ (See in the figure)

On substituting the values of $a,b$ and $m,$ we get

$y=-\sqrt{3}x\pm \sqrt{6\times 3-8}$

$y=-\sqrt{3}x\pm \sqrt{10}$

$y+\sqrt{3}x=\pm \sqrt{10}$

Hence, this is the answer.