If $P$ is $(3, 1)$ and $Q$ is a point on the curve $y^2 = 8x$,then the locus of the mid-point of the line segment $PQ$ is

Suppose that the equation $f(x) = x^{2} + bx + c = 0$ has two distinct real roots $\alpha$ and $\beta$. The angle between the tangent to the curve $y = f(x)$ at the point $\left(\frac{\alpha + \beta}{2}, f\left(\frac{\alpha + \beta}{2}\right)\right)$ and the positive direction of the $x$-axis is (in $^{\circ}$)

What is the vertex of the parabola $x^2 + 4x + 2y - 7 = 0$?

$A$ set of parallel chords of the parabola $y^2 = 4ax$ have their mid-points on:

The directrix of the parabola $2 y^2+25 x=0$ is $........$

Two parabolas with a common vertex at the origin and with axes along the $x-$ axis and $y-$ axis,respectively,intersect each other in the first quadrant. If the length of the latus rectum of each parabola is $3$,then the equation of the common tangent to the two parabolas is?