The focus of a parabolic mirror is at a distance of 8 cm from its center (vertex). If the mirror is 25 cm deep, find the diameter of the mirror.
Let the vertex of the parabolic mirror be at the origin i.e. $V \left(0,0\right)$
Let the equation of the parabolic mirror be: $\;\;$ $y^2 = 4ax$ $\;\;\; \cdots \; (1)$
Given: $\;$ Focus is at a distance of $8$ cm from the center.
$\therefore$ $\;$ $F = \left(a, 0\right) = \left(8, 0\right)$ $\;\;$ $\implies$ $a = 8$ $\;\;\; \cdots \; (2)$
Let the diameter of the parabolic mirror be $AB$.
Depth of the mirror $= 25$ cm
Let $p$ be the radius of the parabolic mirror.
Then, $\;$ $A = \left(25, p\right)$ $\;\;\; \cdots \; (3)$
Now $A$ lies on the parabola.
$\therefore$ $\;$ We have from equations $(1)$, $(2)$ and $(3)$,
$p^2 = 4 \times 8 \times 25 = 800$ $\;\;$ $\implies$ $p = 20 \sqrt{2}$
$\therefore$ $\;$ Diameter $AB = 2p = 40 \sqrt{2}$ cm