The line $y=6x+1$ touches the parabola $y^2=24x.$ The coordinate of a point ‘P’ on this line from which the tangent to $y^2=24x$ is perpendicular to the line $y=6x+1$ is

$y^2=24x\to \left(1\right)\&y=6x+1\to \left(2\right)$

4a = 24

a = 6

 Eq. Of direction is $x=-a$

$\Rightarrow x=-6\to \left(3\right)$

The locus of the P.I of $\text{'}{\perp }^{er\text{'}}$ tangent to a parabola is its directrix

Solve (2) & (3)$\Rightarrow x=-6,y=-35$

Its coordinates are (-6, -35)