Question

Aramis is adjusting a satellite because he finds it is not focusing the incoming radio waves perfectly. The shape of his satellite can be modeled by mc020-1.jpg, where x and y are modeled in inches. He realizes that the static is a result of the feed antenna shifting slightly off the focus point. Where should the feed antenna be placed?
0.75 in. below the vertex
0.75 in. to the right of the vertex
0.75 in. to the left of the vertex
0.75 in. above the vertex

Answer 1

From the picture the parabola passes through (3,3) and has vertex at the origin, so its equation is y = ax^2 with 3 = a(9) ⇒ a = 1/3, i.e. y = (1/3)x^2.

For a parabola in the form y = (1/(4p))x^2 the focus is (0,p). Here 1/(4p) = 1/3 ⇒ 4p = 3 ⇒ p = 3/4 = 0.75.

Thus the focus is at (0,0.75) — 0.75 in. above the vertex.