How do you represent the equation of a circle with a radius of 5 centered at (0,0)?

The equation of a circle in a Cartesian coordinate system is derived from the standard form, which is given by ((x - h)^2 + (y - k)^2 = r^2), where ((h, k)) represents the center of the circle, and (r) is its radius.

In this case, the circle is centered at the origin ((0, 0)) and has a radius of 5. Substituting these values into the standard form of the circle's equation yields:

[

(x - 0)^2 + (y - 0)^2 = 5^2

]

This simplifies to:

[

x^2 + y^2 = 25

]

This matches option B, making it the correct representation of a circle with a radius of 5 centered at the origin. The numerical value of 25 comes from squaring the radius of 5. This emphasizes that the sum of the squares of the distances from any point on the circle to the center remains constant and equals the square of the radius.