Question

If a parabolic reflector is $20cm$ in diameter and $5cm$ deep, Find the focus

Answer 1

Let $F(a,0)$ be the coordinates of the focus of the parabola reflector.

Let $y^2=4ax$ be the equation of the parabola. $\to \left(1\right)$

$\therefore OP=5$

The coordinates of $P=(5,0)$

Since the parabola is symmetric along the x-axis

$AP=PB=\frac{1}{2}AB=\frac{1}{2}\times 20=10$cm $[AB=20$ Given$]$

$\therefore$ Coordinates of $A=(5,10)$

Since A lie on the parabola, substituting $x=5,y=10$ in $\left(1\right)$

$(10{)}^2=4a(5)$

$100=20a$

$a=\frac{100}{20}$

$\therefore a=5$

$\therefore$Focus$=(5,0)$.