Question

If a parabolic reflector is $20\text{cm}$ in diameter and $5\text{cm}$ deep, find the focus.

Answer 1

Finding co-ordinates $A$ and $B$
Let, the equation of parabola is $y^2=4ax$

Given that the mirror is $5\text{cm}$ deep
Hence $x$ − coordinate of point $A\text{and}B\text{is}5$
$\therefore x=5$


From the figure, we can observe that,
$AB=20\text{cm}$
Since parabola is symmetric about $x$ − axis

$\therefore AC=\frac{1}{2}\times AB=10\text{cm}$

Hence coordinates of point $A$ and $B\text{is}(5,10)\text{and}(5,-10)$ respectively.

Finding focus

Since point $A(5,10)$ lies on the parabola, it will satisfy the equation of the parabola

$\Rightarrow y^2=4ax$

$\Rightarrow {10}^2=4\times a\times 5$

$\Rightarrow 100=20a$

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

$\Rightarrow a=5$

$\therefore$ Focus is $F(5,0)$.