Find the equation of the chord of contact of tangents from the point $\left(5,3\right)$ to the hyperbola $4x^2 - 6y^2 = 24$.
Given equation of hyperbola is $\;$ $4x^2 - 6y^2 = 24$
i.e. $\;$ $\dfrac{x^2}{24 / 4} - \dfrac{y^2}{24 / 6} = 1$
i.e. $\;$ $\dfrac{x^2}{6} - \dfrac{y^2}{4} = 1$
The transverse axis of the given hyperbola is along the X axis.
Comparing with the standard equation of the hyperbola $\;$ $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ $\;$ gives
$a^2 = 6$ $\;$ and $\;$ $b^2 = 4$
Equation of chord of contact of tangents from the point $\left(x_1, y_1\right)$ to the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ is: $\;$ $\dfrac{xx_1}{a^2} - \dfrac{yy_1}{b^2} = 1$
Here $\;$ $\left(x_1, y_1\right) = \left(5,3\right)$
$\therefore$ $\;$ The required equation is
$\dfrac{5x}{6} - \dfrac{3y}{4} = 1$
i.e. $\;$ $10x - 9y - 12 = 0$