The equation of the tangent at a point $P(t)$,where $t$ is any parameter,to the parabola $y^2 = 4ax$ is:

$A$ ray of light travelling along the line $x = 2$ strikes a parabolic mirror with the equation $y^2 - 2y - 4x + 5 = 0$ and gets reflected from its surface. Then,the equation of the reflected ray may be:

Find the length of the chord intercepted by the line $4y = 3x - 48$ on the parabola $y^{2} = 64x$.

What is the length of the latus rectum of the parabola $x^2 = -y$?

Find the parametric coordinates of any point on the parabola whose focus is $(0, 1)$ and directrix is $x + 2 = 0$.

The focus of a parabolic mirror as shown in the figure is at a distance of $5 \, cm$ from its vertex. If the mirror is $45 \, cm$ deep,find the distance $AB$. (in $, cm$)