The locus of the poles of focal chords of a parabola is:

The minimum distance between the parabola $y^2 = 8x$ and its image with respect to the line $x + y + 4 = 0$ is:

The line $y = mx + 1$ is a tangent to the parabola $y^2 = 4x$,if

Let $P$ be the point on the parabola ${y^2} = 8x$ which is at a minimum distance from the centre $C$ of the circle ${x^2} + {(y + 6)^2} = 1$. Then the equation of the circle,passing through $C$ and having its centre at $P$,is:

$\triangle OAB$ is an equilateral triangle inscribed in the parabola $y^2 = 4ax, a > 0$ with $O$ as the vertex. Then the length of the side of $\triangle OAB$ is

Let a focal chord $12x + 5y - 27 = 0$ of the parabola $y^2 = kx$ intersect the parabola at the points $P$ and $P^{\prime}$. If $S$ is the focus of this parabola,then $9(SP + SP^{\prime}) = $