If the vertex of the parabola $y = x^2 - 8x + c$ lies on the $x$-axis,then the value of $c$ is:

Find the locus of the point of intersection of perpendicular tangents to the parabola $y^2 - 6y + 24x - 63 = 0$.

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 $\frac{1}{m_1} + \frac{1}{m_2}$?

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

$A$ parabola passing through the point $(-4, -2)$ has its vertex at the origin and the $y$-axis as its axis. The length of the latus rectum of the parabola is:

What is the area of the triangle formed by the vertex of the parabola $x^2 = 12y$ and the endpoints of its latus rectum (in square units)?