Find the tangent to the parabola $\;$ $y^2 = 16x$ $\;$ making an angle of $\;$ $45^\circ$ $\;$ with the $X$ axis.
Let the equation of the tangent be
$y = mx + \dfrac{a}{m}$ $\;\;\; \cdots \; (1)$ $\;\;\;$ where $m$ is the slope of the tangent
The tangent makes an angle of $45^\circ$ with the $X$ axis.
$\therefore \;$ $m = \tan 45^\circ = 1$
Given: $\;$ Equation of parabola: $\;\;\;$ $y^2 = 16x$ $\;\;\; \cdots \; (2)$
Comparing equation $(2)$ with the standard equation of parabola $\;$ $y^2 = 4ax$ $\;$ gives
$4a = 16$ $\implies$ $a = 4$
Substituting the values of $m$ and $a$ in equation $(1)$ gives
$y = 1 \times x + \dfrac{4}{1}$
i.e. $\;$ $y = x + 4$ $\;\;\; \cdots \; (3)$
Equation $(3)$ is the required equation of tangent.