If $ax + by = 1$ is a normal to the parabola $y^2 = 4px$, then the condition is:

The locus of the feet of the perpendiculars drawn from the vertex of the parabola $y^2 = 4ax$ upon all such chords of the parabola which subtend a right angle at the vertex is

If $m_1$ and $m_2$ are the slopes of the tangents drawn from the point $(2, 3)$ to the parabola $y^2 = 4x$,then what is the value of $\frac{1}{m_1} + \frac{1}{m_2}$?

If two parabolas $y^2 = 4x$ and $x^2 = 32y$ intersect at the point $(16, 8)$ at an angle $\theta$,then what is the value of $\tan \theta$?

Let $A$ be the focus of the parabola $y^{2}=8x$. Let the line $y=mx+c$ intersect the parabola at two distinct points $B$ and $C$. If the centroid of the triangle $ABC$ is $(\frac{7}{3},\frac{4}{3})$, then $(BC)^{2}$ is equal to:

Let $P$ be the mid-point of a chord joining the vertex of the parabola $y^{2}=8x$ to another point on it. Then, the locus of $P$ is