Question

this circle is centered at the origin, and the length of its radius is 8. what is the equation of the circle?
a. ( x^{2}+y^{2}=8^{2} )
b. ( \frac{x^{2}}{3}+\frac{y^{2}}{5}=1 )
c. ( x^{2}+y^{2}=8 )
d. ( (x - 8)^{2}+(y - 8)^{2}=64 )

Explanation:

Step1: Recall the standard equation of a circle

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

Step2: Identify the center and radius

Here, the center of the circle is at the origin \((0,0)\) (so \(h = 0\) and \(k = 0\)), and the radius \(r = 8\).

Step3: Substitute values into the standard equation

Substituting \(h = 0\), \(k = 0\), and \(r = 8\) into \((x - h)^{2}+(y - k)^{2}=r^{2}\), we get \((x - 0)^{2}+(y - 0)^{2}=8^{2}\), which simplifies to \(x^{2}+y^{2}=8^{2}\).

Answer:

A. \(x^{2}+y^{2}=8^{2}\)