Two tangents are drawn from a point $(-2, -1)$ to the curve $y^2 = 4x$. If $\alpha$ is the angle between them,then $|\tan \alpha|$ is equal to:

$AB$ is a chord of a parabola $y^2 = 4ax, (a > 0)$ with vertex $A$. $BC$ is drawn perpendicular to $AB$ meeting the axis at $C$. The projection of $BC$ on the axis of the parabola is

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

The equation of the tangent to the parabola $y^2=12x$ at the point $(3,-6)$ is:

The equation of the parabola whose axis is parallel to the $X$-axis and which passes through the points $(-2, 1)$,$(1, 2)$,and $(-1, 3)$ is

If a point $P$ moves such that its distances from the point $A(1, 1)$ and the line $x+y+2=0$ are equal,then the locus of $P$ is