Let $P(2,4)$ and $Q(18,-12)$ be the points on the parabola $y^2=8x$. The equation of the straight line having slope $\frac{1}{2}$ and passing through the point of intersection of the tangents to the parabola drawn at the points $P$ and $Q$ is

Find the locus of the point of contact of the tangent to the parabola $y^2 = 4x$,where the tangent makes an angle of $45^{\circ}$ with the $x$-axis.

If $x = my + c$ is a normal to the parabola ${x^2} = 4ay$,then the value of $c$ is

From the point $(-1, 2)$, tangents are drawn to the parabola $y^2 = 4x$. The area of the triangle formed by the chord of contact and the tangents is: (in $\sqrt{2}$)

$A$ line passing through the point of intersection of $x+y=4$ and $x-y=2$ makes an angle $\tan^{-1}\left(\frac{3}{4}\right)$ with the $X$-axis. It intersects the parabola $y^{2}=4(x-3)$ at points $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$,respectively. Then $|x_{1}-x_{2}|$ is equal to

Find the equation of the normal to the parabola $y^2 = 4x$ which is parallel to the line $y = 3x + 4$.