Question
A parabolic dish antenna has a depth of cm and a diameter of cm at the rim. Where should the receiver be placed so that all incoming parallel rays converge there?
Solution — Step by Step
Place the vertex at origin and the axis of symmetry along the positive -axis. The parabola opens upward: , where is the focal distance (vertex to focus).
The rim is at depth cm and diameter cm. So at , (radius cm).
The focus is at cm — exactly at the depth of the rim, on the axis of symmetry.
Place the receiver at the focus, cm from the vertex along the axis. Rays parallel to the axis reflect through the focus by the optical property of parabolas.
Final answer: receiver at the focus, cm from the vertex along the axis of the dish.
Why This Works
Every parabola has a reflective property — rays parallel to the axis reflect through the focus, and conversely, rays from the focus reflect parallel to the axis. This is why parabolic dishes are used for satellite TV (focus collects signal) and headlights (focus emits light, reflects parallel).
The equation encodes this: is the distance from vertex to focus.
Alternative Method
Use the directrix-focus property. For any point on the parabola, distance from focus = distance from directrix. Plug in the rim point and solve for . Same answer.
For real-world parabola problems: identify “depth” with value at rim, “radius” with value at rim, and use . The focus is always at when the vertex is at origin.
Students sometimes use the diameter as in the equation instead of the radius. Diameter is cm; only half of it ( cm) is the -coordinate of one rim point.