Question

parabolic reflectors
in flashlights, the reflector that directs the light from the bulb is parabolic. if the center of the bulb is placed at the focus of the reflector, then all the generated light will be directed forward in a concentrated beam. in a parabolic reflector, taking a cross section of the reflector through its central axis will create a parabola as shown in the diagram.
part a
a manufacturer is designing a flashlight. for the flashlight to emit a focused beam, the bulb needs to be on the central axis of the parabolic reflector, 3 centimeters from the vertex. write an equation that models the parabola formed when a cross section is taken through the reflectors central axis. assume that the vertex of the parabola is at the origin in the xy - coordinate plane and the parabola can open in any direction.

Explanation:

Step1: Recall parabola equation form

The standard - form equation of a parabola with vertex at the origin \((0,0)\) and opening along the x - axis is \(y^{2}=4px\) (opens to the right if \(p>0\) and to the left if \(p < 0\)), and for a parabola opening along the y - axis is \(x^{2}=4py\) (opens upwards if \(p>0\) and downwards if \(p < 0\)). The distance from the vertex to the focus is \(p\).

Step2: Determine the value of \(p\)

Given that the bulb (at the focus) is 3 centimeters from the vertex. So, \(p = 3\). Since the light is directed forward, assuming the parabola opens to the right (a common orientation for a flashlight reflector), the equation of the parabola is \(y^{2}=4px\).

Step3: Substitute \(p\) into the equation

Substitute \(p = 3\) into \(y^{2}=4px\), we get \(y^{2}=12x\).

Answer:

\(y^{2}=12x\)