Question

Gabe is designing a flashlight that uses a parabolic reflecting mirror and a light source. The shape of the mirror can be modeled by a parabola that opens upward with a diameter of 4 in. and a depth of 1 in. The vertex of the parabola is at the origin. Where should Gabe place the bulb to ensure a perfect beam of light?
(0, 1)
(0, 4)
(1, 0)
(4, 0)

Answer 1

Place the bulb at the focus of the parabola.

Let the parabola be y = ax^2. It passes through (2,1) (half the 4 in diameter), so 1 = a(2)^2 → a = 1/4. Thus y = (1/4)x^2, which is x^2 = 4y, so 4p = 4 and p = 1. The focus is at (0,1).