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 (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?

A. 0.75 in. below the vertex
B. 0.75 in. to the right of the vertex
C. 0.75 in. to the left of the vertex
D. 0.75 in. above the vertex

Answer 1

The correct option is D) 0.75 in. above the vertex.

Explanation:

Consider the provided function.

`(x-4)^2=3(y-3)`

The general equation of the parabola is:

`(x-x_0)^2=4p(y-y_0)`

Where, p is the focus of the parabola and (x₀,y₀) is the vertex of the parabola .

Now by comparing we can concluded that:

The vertex of the parabola is (4,3)

`3=4p\\ p=(3)/(4)\\ p=0.75`

We have to shift the antenna towards the focus point. The focus point would be: (x₀,y₀+p)=(4,3+0.75)= (4,3.75).

Hence, the correct option is D) 0.75 in. above the vertex.

Answer 2

Given the equation of the parabola

`(x-4)^2=3(y-3).`

The vertex of this parabola is placed at point (4,3).

If the equation of the parabola is
`(x-x_0)^2=2p(y-y_0),` then

`2p=3,\\ \\p=1.5.`

The coordinates of the parabola focus are

`\left(x_0,y_0+(p)/(2)\right).`

Therefore, the focus is placed at point (4,3,75).

Answer: option D, 0.75 in. above the vertex