To determine the radius of a circle given its equation in the form [tex]\(x^2 + y^2 = z\)[/tex], we can compare it to the standard form of a circle's equation.
The standard form of the equation of a circle is:
[tex]\[
(x - h)^2 + (y - k)^2 = r^2
\][/tex]
In this equation, [tex]\((h, k)\)[/tex] represents the coordinates of the center of the circle, and [tex]\(r\)[/tex] represents the radius of the circle.
Given the equation:
[tex]\[
x^2 + y^2 = z
\][/tex]
We can see that this equation is already in the form where the center of the circle, [tex]\((h, k)\)[/tex], is at [tex]\((0, 0)\)[/tex], since there are no shifts in the [tex]\(x\)[/tex] or [tex]\(y\)[/tex] directions (i.e., there are no [tex]\((x-h)\)[/tex] or [tex]\((y-k)\)[/tex] terms). Therefore, we can rewrite the equation as:
[tex]\[
(x - 0)^2 + (y - 0)^2 = z
\][/tex]
Simplifying this, we get:
[tex]\[
x^2 + y^2 = z
\][/tex]
When we compare this to the standard form [tex]\((x - h)^2 + (y - k)^2 = r^2\)[/tex], we can identify that [tex]\(r^2 = z\)[/tex].
To find the radius [tex]\(r\)[/tex], we need to take the square root of both sides of the equation [tex]\(r^2 = z\)[/tex]:
[tex]\[
r = \sqrt{z}
\][/tex]
Therefore, the correct answer is:
The radius is the square root of the constant term, [tex]\(z\)[/tex].
