Give the focus, directrix, and axis for the parabola.
-A domed ceiling is a parabolic surface. For the best lighting on the floor, a light source is to be placed at the focus of the surface. If 10 m down from the top of the dome the ceiling is 19 m wide,
Find the best location for the light source.
A) 9 m down from the top
B) 4.6 m down from the top
C) 2.3 m down from the top
D) 6.9 m down from the top

Question

Quizplus · Accepted Answer

The best location for the light source is at the focus of the parabola. Given the width (19 m) and the depth (10 m) from the top of the dome, we can use the formula for a parabola in vertex form, $y = \frac{1}{4p}x^2$, where $2p$ is the focal length (distance between the vertex and the focus), and the width at the given depth is $2a$, where $a$ is the distance from the vertex to the parabola's edge at that depth. Here, $a = 9.5$ m (half of 19 m), and $y = 10$ m. Solving for $p$ gives us the distance from the vertex to the focus. Since $a^2 = 4py$, plugging in $a = 9.5$ and $y = 10$ gives $p = 2.3$. Therefore, the light source should be placed 2.3 m down from the top of the dome to be at the focus.

Give the Focus, Directrix, and Axis for the Parabola