The equation of the tangent to the parabola $y^2 = 4ax$ at the point $(a/t^2, 2a/t)$ is:

If $x + y = k$ is a normal to the parabola ${y^2} = 12x$,then $k$ is

Let $O$ be the vertex of the parabola $x^{2}=4y$ and $Q$ be any point on it. Let the locus of the point $P$, which divides the line segment $OQ$ internally in the ratio $2:3$, be the conic $C$. Then the equation of the chord of $C$, which is bisected at the point $(1, 2)$, is:

$A$ tangent to the parabola $y^2 = 8x$ makes an angle of $45^\circ$ with the straight line $y = 3x + 5$. Find the equation of the tangent.

If a chord of the parabola $y^2=4x$ passes through its focus and makes an angle $\theta$ with the $X$-axis,then its length is

The focus of the parabola $4y^2 - 6x - 4y = 5$ is