The focus of a parabolic mirror is at a distance of 6 cm from its vertex. If the mirror is 20 cm deep, find its diameter.
Given: The focus of a parabolic mirror is at a distance of 6 cm from its vertex. And the mirror is 20 cm deep.
Need to find: Diameter of the mirror
Here $O$ is the vertex and $A$ is the Focus. So, $O A=a=6 \mathrm{~cm}$.
OD is the deep of the mirror $=20 \mathrm{~cm}$
$B C$ is the diameter of the mirror.
Equation of the parabola is, $\mathrm{y}^{2}=4 \mathrm{ax}$
$\Rightarrow y^{2}=24 x$
The mirror is $20 \mathrm{~cm}$ deep. That means the $x$-coordinate of the points $B, C$ and $D$ is 20
Both the points, B and C are on the parabola. Hence, the points satisfies the equation of the parabola.
Therefore, $y^{2}=24 \times 20=480$
$\Rightarrow y=\pm 21.9$
So, the coordinate of $B$ is $(20,21.9)$ and the coordinate of $C$ is $(20,-21.9)$.
Therefore, the diameter of the mirror is $=(21.9+21.9) \mathrm{cm}$
$=43.8 \mathrm{~cm}$
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