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, [tex]z[/tex].
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 (2)
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 .
Taking the square root of both sides, we find r = z $.
Thus, the radius is the square root of the constant term z .
The radius of the circle is z .
Explanation
Understanding the circle equation 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 . We need to find the relationship between r and z .
Relating z and r^2 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 . Therefore, we have the equation r 2 = z .
Finding the radius To find the radius r , we need to take the square root of both sides of the equation r 2 = z . This gives us r = z $.
Conclusion Therefore, the radius of the circle is the square root of the constant term z .
Examples
Understanding the radius of a circle is crucial in many real-world applications. For example, when designing a circular garden, knowing the radius allows you to calculate the amount of fencing needed or the area available for planting. Similarly, in architecture, the radius is essential for designing circular structures like domes or arches, ensuring structural integrity and aesthetic appeal. In physics, understanding circular motion relies heavily on the concept of radius, such as calculating the speed of an object moving in a circular path. The relationship between the radius and the equation of a circle, x 2 + y 2 = r 2 , provides a fundamental tool for solving various practical problems across different fields.
The radius of a circle given the equation x 2 + y 2 = z is found by taking the square root of the constant term z . Therefore, the correct choice is option C: the radius is the square root of the constant term, z .
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