Question

A radio telescope has a parabolic surface, as shown below.If the telescope is 1 m deep and 8 m wide, how far is the focus from the vertex?

Answer 1

4 m

Explanation:

The equation can be written ...

y = (x/4)² = (1/16)x²

where the divisor 4 is the horizontal scale factor that makes the parabola 8 m wide at y = 1 m.

This can also be written as ...

y = 1/(4p)x² = (1/16)x²

from which we can see ...

4p = 16

p = 4

In this form, p is the distance from the vertex to the focus, 4 meters.

_____

Another way to find the y-coordinate of the focus is to draw a line with slope 1/2 through the vertex. It will intersect the parabola at the point where the vertical distance to the directrix is the same as the horizontal distance to the focus. The y-coordinate of that point is the y-coordinate of the focus.