Find the value of $k$ so that the line $3x - y + k = 0$ may touch the parabola $y^2 = 24x$.
Given parabola: $\;$ $y^2 = 24x$ $\;\;\; \cdots \; (1)$
Given line: $\;$ $3x - y + k = 0$ $\;$ i.e. $\;$ $y = 3x + k$ $\;\;\; \cdots \; (2)$
Substituting the value of $y$ from equation $(2)$ in equation $(1)$ gives
$\left(3x + k\right)^2 = 24x$
i.e. $\;$ $9x^2 + 6kx + k^2 = 24x$
i.e. $\;$ $9x^2 + \left(6k - 24\right) x + k^2 = 0$ $\;\;\; \cdots \; (3)$
Since equation $(2)$ touches the parabola given by equation $(1)$, line $(2)$ meets the parabola in two coincident points.
i.e. $\;$ the discriminant of equation $(3)$ is equal to $0$ (zero).
i.e. $\;$ Discriminant $= \Delta = \left(6k - 24\right)^2 - 4 \times 9 \times k^2 = 0$
i.e. $\;$ $36 k^2 - 288 k + 576 - 36 k^2 = 0$
i.e. $\;$ $288 k = 576$
$\implies$ $k = 2$