Question

A parabolic reflector is 9 cm deep and its diameter is 24 cm. How far is its focus from the vertex?

Answer 1

Let us take the vertex of the reflector at the origin, and let its axis lie along the positive direction of the x-axis.

Let the equation of the parabolic shape of the reflector be

$y^2=4ax....\left(i\right)$

As given, we have

OM = 9 cm and AB = 24 cm

$\Rightarrow$ OM = 9 cm and $AM=\frac{1}{2}$ AB = 12 cm.

$\therefore$ the coordinates of A are (9, 12).

Since A lies on (i), we have

$12\times 12=4\times a\times 9\Rightarrow a=4.$

$\therefore$ the equation of parabola is $y^2=16x.$

Its focus is F(a, 0), i.e., F(4, 0).

Hence, the focus is at a distance of 4 cm from the vertex.