To determine where the feed antenna should be placed, given the equation of the satellite's shape [tex]\((x-4)^2 = 3(y-3)\)[/tex], let's solve the problem step-by-step:
1. Identify the Vertex:
The equation of the parabola is given in the form [tex]\((x - h)^2 = 3(y - k)\)[/tex], where [tex]\((h, k)\)[/tex] represents the vertex of the parabola.
Comparing [tex]\((x - 4)^2 = 3(y - 3)\)[/tex] with the standard form [tex]\((x - h)^2 = 4p(y - k)\)[/tex],
we find that:
[tex]\[
h = 4
\][/tex]
[tex]\[
k = 3
\][/tex]
So, the vertex of the parabola is at [tex]\((4, 3)\)[/tex].
2. Determine the Focus:
The standard form of a vertical parabola is [tex]\((x - h)^2 = 4p(y - k)\)[/tex], where [tex]\(p\)[/tex] represents the distance from the vertex to the focus.
Given the equation [tex]\((x - h)^2 = 3(y - k)\)[/tex], we notice that [tex]\(4p = 3\)[/tex]. Therefore, we can solve for [tex]\(p\)[/tex]:
[tex]\[
4p = 3 \implies p = \frac{3}{4}
\][/tex]
The focus of the parabola is located at [tex]\((h, k + p)\)[/tex].
Hence, the coordinates of the focus are:
[tex]\[
(4, 3 + \frac{3}{4}) = (4, 3.75)
\][/tex]
3. Determine the Correct Placement for the Feed Antenna:
According to the problem, the feed antenna has shifted slightly off the focus point. We need to determine how far and in which direction.
Among the choices given:
- 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
Since the focus of the parabola is at [tex]\((4, 3.75)\)[/tex], to compensate for the misalignment and ensure the optimal concentration of radio waves, the feed antenna should be placed above the vertex.
Specifically, we need to place the feed antenna:
[tex]\[
3 + 0.75 = 3.75 \quad \text{(y-coordinate)}
\][/tex]
Thus, the correct answer is:
[tex]\( 0.75 \text{ in. above the vertex}. \)[/tex]
