To find the coordinates of the focus for the parabola given by the equation [tex]\( x^2 = 8y \)[/tex], follow these steps:
1. Identify the standard form of a parabola that opens upwards: The standard form for this type of parabola is [tex]\( x^2 = 4ay \)[/tex], where [tex]\( a \)[/tex] is the distance from the vertex to the focus.
2. Rewrite the given equation in the standard form: The given equation is [tex]\( x^2 = 8y \)[/tex].
3. Compare the given equation to the standard form: By comparing [tex]\( x^2 = 8y \)[/tex] to [tex]\( x^2 = 4ay \)[/tex], notice that:
[tex]\[
4a = 8
\][/tex]
4. Solve for [tex]\( a \)[/tex]:
[tex]\[
a = \frac{8}{4} = 2
\][/tex]
5. Determine the coordinates of the focus: For a parabola [tex]\( x^2 = 4ay \)[/tex], the focus is located at [tex]\( (0, a) \)[/tex]. Using the value of [tex]\( a \)[/tex] we found:
[tex]\[
a = 2
\][/tex]
Therefore, the coordinates of the focus are [tex]\( (0, 2) \)[/tex].
So, the focus of the parabola described by the equation [tex]\( x^2 = 8y \)[/tex] is at [tex]\( \boxed{(0, 2)} \)[/tex].
