Chapter 8.1, Problem 62E

Solution

To State:

Where should the light bulb be placed inside the given parabolic reflector surface of a headlight.

At the point $\left(0,3\right)$ .

Given:

The parabolic reflector surface of the headlight is generated by rotating the parabola $x^2=12y$ about its axis.

Concepts Used:

For best directionality in a light beam the light source must be placed at the focus of a parabolic reflector surface.

The equation $x^2=4py$ represents a parabola with vertex $\left(0,0\right)$ and focus at $\left(0,p\right)$ .

Calculations:

Rewrite the given equation $x^2=12y$ in the format $x^2=4py$ :

  $\begin{array}{c}{x}^2=12y\\ x^2=4\left(3\right)y\end{array}$

Compare the above result with the equation of standard parabola $x^2=4py$ and write the coordinates of the focus $\left(0,p\right)$ :

  $\left(0,p\right)=\left(0,3\right)$

Conclusion:

The light bulb should be placed at the point $\left(0,3\right)$ inside the given parabolic reflector surface of a headlight.