Difficult parabola question(s)

Question

Premise: The reflector if a satellite dish is in the shape of a parabola with a diameter of 4 feet and a depth of 2 feet. To get the maximum reception we need to replace the antenna at the focus.

a. Write the equation of the parabola of the cross section of the dish, placing the vertex of the parabola at the origin. Convert the equation into standard form, if necessary. What is the defining feature of the equation that tells us it is a parabola?

b. Describe the graph of the parabola. Find the vertex, directrix, and focus.

c. Use the endpoints of the latus rectum to find the focal width.

d. How far above the vertex should the receiving antenna be placed?

Answer 1

General Equation: x^2 = (4p)y

To find p, use the given point (2,2)

(2)^2 = (4p)(2)

4 = 8p

1/2 = p

Equation of Parabola: x^2 = 2y

Vertex at origin

Directrix:

y = -p

y = - 1/2

Focal Width = 4p = 4(1/2) = 2

Receiving Antenna at Focus

Focus at: (x,y) = (0,p) = (0,1/2)

Answer 2

The equation of the general parabola with axis parallel to the y axis and vertex at the origin is:

y=x²/4p where (0,p) is the focus.

The directrix is y=-p and the length of the latus rectum is 4p.

You should be able to make the necessary substitutions.

Note: What is described in the question is actually a circular paraboloid; the cross section is the parabola.