To determine the center of a circle given by the equation [tex]\( (x+9)^2 + (y-6)^2 = 10^2 \)[/tex], we need to compare this equation to the standard form of the equation of a circle.
The standard form of the equation of a circle is:
[tex]\[ (x-h)^2 + (y-k)^2 = r^2 \][/tex]
where [tex]\((h, k)\)[/tex] represents the center of the circle and [tex]\(r\)[/tex] is the radius.
By comparing the given equation [tex]\( (x+9)^2 + (y-6)^2 = 10^2 \)[/tex] with the standard form, we can identify the coordinates of the center [tex]\((h, k)\)[/tex].
1. The term [tex]\( (x+9)^2 \)[/tex] implies that [tex]\( h = -9 \)[/tex]. This is because [tex]\( (x - (-9))^2 = (x + 9)^2 \)[/tex].
2. The term [tex]\( (y-6)^2 \)[/tex] implies that [tex]\( k = 6 \)[/tex]. This is because [tex]\( (y - 6)^2 \)[/tex] is already in the correct format.
Therefore, the center of the circle is:
[tex]\[ (-9, 6) \][/tex]
Given the options:
- [tex]\((-9, 6)\)[/tex]
- [tex]\((-6, 9)\)[/tex]
- [tex]\((6, -9)\)[/tex]
- [tex]\((9, -6)\)[/tex]
The correct answer is:
[tex]\[ (-9, 6) \][/tex]
