The equation of the parabola with focus $(3, 0)$ and the directrix $x + 3 = 0$ is

The shortest distance between the line $y-x=1$ and the curve $x=y^2$ is

Find the equation of the common tangent to the parabolas $y^2 = 2x$ and $x^2 = 16y$.

Tangent and normal are drawn at $P(16, 16)$ on the parabola ${y^2} = 16x$,which intersect the axis of the parabola at $A$ and $B$,respectively. If $C$ is the centre of the circle through the points $P, A$ and $B$ and $\angle CPB = \theta$,then a value of $\tan \theta$ is:

The equation of the directrix of the parabola whose focus is $(0,0)$ and the tangent at the vertex is $x-y+1=0$ is

If $2x + 3y + 12 = 0$ and $x - y + 4\lambda = 0$ are conjugate with respect to the parabola $y^2 = 8x$,then $\lambda$ is equal to