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A parabolic reflector is 9 cm deep and i...

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

Text Solution

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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.` ....(i)
As given, we have
OM = 9 cm and AB = 24 cm
` rArr OM = 9 cm and AM = 1/2 AB = 12 cm.`
` :. ` the coordinates of A are (9, 12).
Since A lies on (i), we have
` 12 xx 12=4xx a xx 9 rArr a = 4.`
`:. ` the equation of the 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.
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