Question

which of the following equations correctly represents a circle centered at the origin with a radius of 5?
a. $(x - 5)^2+(y - 5)^2 = 5$
b. $(x - 5)^2+y^2 = 25$
c. $x^2+y^2 = 25$
d. $x^2+y^2 = 5$

Explanation:

Step1: Recall the standard equation of a circle

The standard equation of a circle is \((x - h)^{2}+(y - k)^{2}=r^{2}\), where \((h,k)\) is the center and \(r\) is the radius.

Step2: Substitute the center and radius values

Since the center is at the origin \((0,0)\) (so \(h = 0,k = 0\)) and radius \(r = 5\), substituting into the formula gives \((x - 0)^{2}+(y - 0)^{2}=5^{2}\), which simplifies to \(x^{2}+y^{2}=25\).

Answer:

C. \(x^{2}+y^{2}=25\)