Question

If a parabola reflector with its axis as $x-axis$ is $20$ cm in diameter and $5$ cm deep. Find the focus.

Answer 1

The origin of the coordinate plane is taken at the vertex of the parabolic reflector in such a way that the axis of the reflector is along the positive $x$-axis.
The equation of the parabola is of the form $y^2=4ax$ (as it is opening to the right)

Given, diameter of parabola $=20$ cm

Since, parabola is symmetric about $x$-axis.

So, co-ordinates of point $A$ are $(5,10)$
Since the parabola passes through the point $A(5,10)$

${10}^2=4a\left(5\right)$
$\Rightarrow 100=20a$
$\Rightarrow a=\frac{100}{20}=5$
Therefore, the focus of the parabola is $(a,0)=(5,0)$ which is the mid-point of the diameter.