Which of the following points lie on the parabola $x^2 = 4ay$?

The coordinates of the extremities of the latus rectum of the parabola $5y^2 = 4x$ are

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 $^{\circ}$)

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

If the tangents and normals at the extremities of a focal chord of a parabola $y^2 = 4ax$ intersect at $(x_1, y_1)$ and $(x_2, y_2)$ respectively,then:

Let $(x, y)$ be any point on the parabola $y^2 = 4x$. Let $P$ be the point that divides the line segment from $(0, 0)$ to $(x, y)$ in the ratio $1:3$. Then the locus of $P$ is