If the eccentricity of the standard hyperbola passing through the...

Let us suppose equation of hyperbola is
$\dfrac{{x}^{2}}{{a}^{2}} - \dfrac{{y}^{2}}{{b}^{2}} = 1$
$e = 2$

$∵ {b}^{2} = \left({e}^{2} - 1\right){a}^{2}$

$⇒ {b}^{2} = 3{a}^{2}$
It is passing through \left(4,6\right)$</p><p>$\dfrac{16}{{a}^{2}} - \dfrac{36}{{b}^{2}} = 1$</p><p>$⇒ {a}^{2} = 4, {b}^{2} = 12$<br />So, equation of tangent is$T = 0$</p><p>$\dfrac{x{x}_{1}}{{a}^{2}} - \dfrac{y{y}_{1}}{{b}^{2}} = 1$</p><p>Tangent at point$( 4, 6 )$is<br />$⇒ x - \dfrac{y}{2} = 1$<br />$⇒ 2x - y - 2 = 0$