To solve this problem, we need to understand the standard form of the equation of a circle. The general form of a circle's equation centered at the origin is:
[tex]\[ x^2 + y^2 = r^2 \][/tex]
Here, [tex]\( r \)[/tex] represents the radius of the circle.
Given the equation:
[tex]\[ x^2 + y^2 = z \][/tex]
you can see that the equation is in the form [tex]\( x^2 + y^2 = r^2 \)[/tex]. By comparing both equations, we can directly infer that the constant term [tex]\( z \)[/tex] on the right side of the equation is equal to [tex]\( r^2 \)[/tex], the square of the radius.
To find the radius [tex]\( r \)[/tex], we need to take the square root of the constant term [tex]\( z \)[/tex]:
[tex]\[ r = \sqrt{z} \][/tex]
Thus, the radius is the square root of the constant term [tex]\( z \)[/tex].
Given a specific example where [tex]\( z = 25 \)[/tex]:
1. We recognize that the equation is [tex]\( x^2 + y^2 = 25 \)[/tex].
2. Here, [tex]\( z = 25 \)[/tex].
3. The radius [tex]\( r \)[/tex] is then [tex]\( \sqrt{25} \)[/tex].
4. Calculating [tex]\( \sqrt{25} \)[/tex] gives us [tex]\( 5 \)[/tex].
So, the radius of the circle is [tex]\( 5 \)[/tex].
Therefore, the correct answer to the problem is:
[tex]\[ \text{The radius is the square root of the constant term, } z. \][/tex]
