You are designing a lamp reflector for a spotlight.

Question

You are designing a lamp reflector for a spotlight. If you want to placc the lamp 50cm from the vertex of the parabolic reflector, find the equation of the parabola that describes the shape of the reflector.

Answer 1

For the best reflector, the lamp should be placed at the focus of the parabola which is 50 cm.

Draw a vertical parabola opening up with the vertex at the origin. So, y = ax². The focus is directly above the vertex at (0,1/(4a)).

50 = 1/(4a) --> a = 1/200

y = x²/200

Answer 2

x² = 4py; p = 50. So, x² = 200y (x, y in cm)