To determine which equation represents Circle Q, we should recall the standard form of the equation for a circle centered at the origin [tex]\((0, 0)\)[/tex] with radius [tex]\(r\)[/tex]. This general equation is:
[tex]\[ x^2 + y^2 = r^2 \][/tex]
Given that Circle Q has a radius of 9 units, we need to substitute [tex]\(r\)[/tex] with 9 in the standard equation:
[tex]\[ x^2 + y^2 = 9^2 \][/tex]
Next, calculate [tex]\(9^2\)[/tex]:
[tex]\[ 9^2 = 81 \][/tex]
So, substituting it back into the equation:
[tex]\[ x^2 + y^2 = 81 \][/tex]
Therefore, the equation that represents Circle Q is:
[tex]\[ x^2 + y^2 = 81 \][/tex]
Among the given options:
- [tex]\( x^2 + y^2 = 3 \)[/tex] is incorrect as it implies a radius of [tex]\(\sqrt{3}\)[/tex].
- [tex]\( x^2 + y^2 = 9 \)[/tex] is incorrect as it implies a radius of 3.
- [tex]\( (x-9)^2 + (y-9)^2 = 1 \)[/tex] is incorrect as it represents a circle centered at [tex]\((9, 9)\)[/tex] with a radius of 1.
- [tex]\( x^2 + y^2 = 81 \)[/tex] is correct.
Thus, the correct equation that represents Circle Q is:
[tex]\[ x^2 + y^2 = 81 \][/tex]
