Which explains how to find the radius of a circle whose equation is in the form [tex]x^2+y^2=z[/tex]?
A. The radius is the constant term, z.
B. The radius is the constant term, [tex]z[/tex], divided by 2.
C. The radius is the square root of the constant term, [tex]z[/tex].
D. The radius is the square of the constant term, [tex]z[/tex].
Answer (1)
The equation of a circle centered at the origin is x 2 + y 2 = r 2 , where r is the radius.
Given the equation x 2 + y 2 = z , we equate z to r 2 .
To find the radius r , we take the square root of z : r = z $
.
Therefore, the radius is the square root of the constant term z : z .
Explanation
Recall the standard equation of a circle. The equation of a circle centered at the origin is given by x 2 + y 2 = r 2 , where r is the radius of the circle. We are given the equation x 2 + y 2 = z .
Compare the given equation with the standard equation. Comparing the given equation x 2 + y 2 = z with the standard equation x 2 + y 2 = r 2 , we can see that z corresponds to r 2 .
Solve for the radius. To find the radius r , we need to take the square root of z . Therefore, r = z $
.
State the correct method. Thus, the radius is the square root of the constant term z .
Examples
Imagine you're designing a circular garden and you know the area you want the garden to cover. If the equation representing the garden's boundary is x 2 + y 2 = 25 , then z = 25 . To find the radius of the garden, you would take the square root of 25, which is 5. So, the radius of your circular garden would be 5 units. This helps you determine how much fencing you need or how to arrange plants from the center to the edge.
