Franco 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 y^2+6y-3x+3=0 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. What is the focus point of the satellite?

We have to find the focus of the parabola with equation:

[tex]y^{2}+6y-3x+3=0[/tex]

In order to find the focus, we first need to convert the equation of parabola to its standard form, as shown below:

[tex]y^{2} + 6y = 3x-3 \\ \\ y^{2} + 2(y)(3) = 3x-3 \\ \\ y^{2} + 2(y)(3) + (3)^{2} = 3x-3 + (3)^{2} \\ \\ (y + 3)^{2} = 3x + 6 \\ \\ (y+3)^{2} = 3(x+2) \\ \\ (y+3)^{2} = 4 * \frac{3}{4} (x+2)[/tex]

Comparing this equation to the similar general equation of parabola, we get:

[tex](y-k)^{2} = 4p(x-h)[/tex]

k = -3
p= 3/4
h = - 2

The focus of the general parabola is located at ( h+p, k)

Using the values, we get:

Focus of the given parabolic satellite = [tex](-2+ \frac{3}{4},-3)=(- \frac{5}{4},-3) [/tex]