The equation of a tangent to the parabola, x ^ 2 = 8y , which makes...

As we know, if the line $y = mx + c$ is a tangent to ${x}^{2} = 8y = 4 \times 2 \times y$, then $c = - 2{m}^{2}$.

So, equation of tangent with slope $m$ is

$y = mx - 2{m}^{2} . . . \left(i\right)$

Since, tangent is inclined at $\theta$ with the positive $x -$ axis, i.e. $m = tan \theta$.

Equation of tangent is $y = tan\theta . x - 2\tan^{2} \theta$

$⇒ ycot \theta = x - 2tan\theta$

$⇒ x = ycot\theta + 2tan\theta$