To determine the coordinates of the focus of the given parabola [tex]\(x^2 = -20y\)[/tex], we should first compare it to the standard form of a parabola that opens either upwards or downwards, which is [tex]\(x^2 = 4py\)[/tex].
1. Comparing the Standard Form:
The given equation [tex]\(x^2 = -20y\)[/tex] can be compared to the standard form [tex]\(x^2 = 4py\)[/tex].
2. Identify [tex]\(4p\)[/tex]:
From the comparison, we can see that [tex]\(4p = -20\)[/tex].
3. Solve for [tex]\(p\)[/tex]:
To find [tex]\(p\)[/tex], we solve the equation [tex]\(4p = -20\)[/tex]:
[tex]\[
p = \frac{-20}{4} = -5
\][/tex]
4. Determine the Focus:
The focus of a parabola [tex]\(x^2 = 4py\)[/tex] is located at [tex]\((0, p)\)[/tex]. From the calculation above, we have [tex]\(p = -5\)[/tex].
Therefore, the coordinates of the focus are [tex]\((0, -5)\)[/tex].
Among the given options:
- [tex]\((-5,0)\)[/tex]
- [tex]\((5,0)\)[/tex]
- [tex]\((0,5)\)[/tex]
- [tex]\((0,-5)\)[/tex]
The correct coordinates of the focus are [tex]\((0, -5)\)[/tex].
