Show that the line $12y - 20x - 9 = 0$ touches the parabola $y^2 = 5x$.
Equation of parabola: $\;$ $y^2 = 5x$ $\;\;\; \cdots \; (1)$
Equation of line: $\;$ $12y - 20x - 9 = 0$ $\;$ i.e. $\;$ $y = \dfrac{20x + 9}{12}$ $\;\;\; \cdots \; (2)$
In view of equation $(2)$, equation $(1)$ becomes
$\left(\dfrac{5}{3} x + \dfrac{3}{4}\right)^2 = 5x$
i.e. $\;$ $\dfrac{25}{9} x^2 + \dfrac{9}{16} + \dfrac{5}{2} x = 5x$
i.e. $\;$ $\dfrac{25}{9} x^2 - \dfrac{5}{2} x + \dfrac{9}{16} = 0$ $\;\;\; \cdots \; (3)$
Discriminant of equation $(3)$ is
$\Delta = \left(\dfrac{-5}{2}\right)^2 - 4 \times \dfrac{25}{9} \times \dfrac{9}{16} = \dfrac{25}{4} - \dfrac{25}{4} = 0$
$\implies$ Roots of equation $(3)$ are equal.
$\implies$ Equation $(2)$ touches equation $(1)$
i.e. $\;$ equation $(2)$ is a tangent to the given parabola.