To determine the coordinates of the focus for the satellite dish described by the equation [tex]\(x^2 = 8y\)[/tex], we need to recognize that this is a parabolic equation in standard form. For a parabola that opens upwards and has its vertex at the origin [tex]\((0,0)\)[/tex], the standard equation is given by:
[tex]\[ x^2 = 4py \][/tex]
Where [tex]\((0, p)\)[/tex] is the focus of the parabola.
1. Identify the given equation and compare it with the standard form:
The given equation is:
[tex]\[ x^2 = 8y \][/tex]
2. Match the given equation to the standard form [tex]\(x^2 = 4py\)[/tex]:
[tex]\[ x^2 = 4py \][/tex]
By comparing it with:
[tex]\[ x^2 = 8y \][/tex]
We see that [tex]\(4p = 8\)[/tex].
3. Solve for [tex]\(p\)[/tex]:
[tex]\[ 4p = 8 \][/tex]
[tex]\[ p = \frac{8}{4} \][/tex]
[tex]\[ p = 2 \][/tex]
4. Determine the coordinates of the focus:
Since the focus is located at [tex]\((0, p)\)[/tex], we substitute [tex]\(p = 2\)[/tex] into the coordinates.
Thus, the coordinates of the focus are [tex]\((0, 2)\)[/tex].
Therefore, the coordinates of the focus of the satellite dish described by the equation [tex]\(x^2 = 8y\)[/tex] are:
[tex]\[
\boxed{(0, 2)}
\][/tex]
