Question

aramis is adjusting a satellite because he finds it is not focusing the incoming radio waves perfectly. the shape of the satellite can be modeled by $(x - 4)^2 = 3(y - 3)$, 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

Explanation:

Step1: Recall the standard form of a parabola

The standard form of a parabola is \((x - h)^2=4p(y - k)\), where \((h,k)\) is the vertex and \(p\) is the distance from the vertex to the focus.
Given the equation \((x - 4)^2=3(y - 3)\), we can rewrite it as \((x - 4)^2 = 4\times\frac{3}{4}(y - 3)\).

Step2: Identify the vertex and the value of \(p\)

For the equation \((x - 4)^2=3(y - 3)\), the vertex is \((h,k)=(4,3)\).
Comparing with \((x - h)^2 = 4p(y - k)\), we have \(4p = 3\), so \(p=\frac{3}{4}=0.75\).
Since the parabola is of the form \((x - h)^2=4p(y - k)\) (opens up - ward), the focus is at \((h,k + p)\).

Answer:

\(0.75\) in. above the vertex