Question

For all real permissible values of $m,$ if the straight line $y=mx+\sqrt{9m^2-4}$ is tangent to a hyperbola, then equation of the hyperbola can be

• A. $9x^2-4y^2=64$
• B. $4x^2-9y^2=64$
• C. $9x^2-4y^2=36$
• D. $4x^2-9y^2=36$

Answer 1

The correct option is D $4x^2-9y^2=36$

Let the equation of tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ is
$y=mx\pm \sqrt{a^2{m}^2-b^2}$
Comparing it with $y=mx+\sqrt{9m^2-4},$ we have
$a^2=9$ and $b^2=4$
$\therefore$ Equation of the hyperbola is :
$\frac{x^2}{9}-\frac{y^2}{4}=1$
$\Rightarrow 4x^2-9y^2=36$