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)$<br/><br/>$⇒ 4a^{2} - b^{2} = 16$...(ii)<br/><br/>On solving Eqs. (i) and (ii), we get,<br/><br/>$a^{2} = 5, b^{2} = 4$<br/><br/>$∴$Equation of hyperbola is \dfrac{x^{2}}{5}$– $\dfrac{y^{2}}{4} =$1$<br/><br/>$⇒ 4x^{2} - 5y^{2} = 20