In which quadrant does the vertex of the parabola $y^2 + 2y + x = 0$ lie?

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

If the line $lx + my + n = 0$ is a tangent to the parabola $y^2 = 4ax$,then the locus of its point of contact is

Suppose $AB$ is a focal chord of the parabola $y^2=12x$ of length $l$ and slope $m < \sqrt{3}$. If the distance of the chord $AB$ from the origin is $d$,then $l \cdot d^2$ is equal to ....................

The condition that the line $ax + by + c = 0$ is tangent to the parabola $y^2 = 4ax$ is:

If the points of intersection of the parabolas $y^2=5x$ and $x^2=5y$ lie on the line $L$,then the area of the triangle formed by the directrix of one parabola,the latus rectum of another parabola,and the line $L$ is