Question

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.

Answer 1

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, OA = a = 6 cm. 

OD is the deep of the mirror = 20 cm 

BC is the diameter of the mirror. 

Equation of the parabola is, y² = 4ax 

⇒ y² = 24x 

The mirror is 20 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, y² = 24 x 20 = 480 

⇒ y = ± 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) cm 

= 43.8 cm