If the focus of a hyperbola is ( ± 3,0 ) and the equation of a...

Given, $\left(\pm ae, 0\right) = \left(\pm3, 0\right)<br /><br />$⇒ ae = 3$<br /><br />$⇒ a^{2}e^{2} = 9$<br /><br />${⇒ b}^{2} + a^{2} = 9$...(i)<br /><br />$∵ 2x + y - 4 = 0$<br /><br />$⇒ y = - 2x + 4$<br /><br />is the tangent to the hyperbola<br /><br />$∴ \left(4\right)2 = a^{2} \left(-2\right)$2 - b^{2}\left(∵ c^{2} = a^{2}m^{2} - b^{2}\right)$

$⇒ 4a^{2} - b^{2} = 16$ ...(ii)

On solving Eqs. (i) and (ii), we get,

$a^{2} = 5, b^{2} = 4$

$∴$ Equation of hyperbola is $\dfrac{x^{2}}{5}– \dfrac{y^{2}}{4} =1$

$⇒ 4x^{2} - 5y^{2} = 20$