Question

which of the following equations correctly represents a circle centered at the origin with a radius of 10?
a. $x^{2}+y^{2}=100$
b. $x^{2}+y^{2}=100^{2}$
c. $(x - 10)^{2}+(y - 10)^{2}=100$
d. $x^{2}+y^{2}=10$

Explanation:

Step1: Recall circle - equation formula

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

Step2: Identify center and radius values

Since the circle is centered at the origin \((0,0)\), \(h = 0\) and \(k = 0\), and the radius \(r = 10\).

Step3: Substitute values into formula

Substituting \(h = 0\), \(k = 0\), and \(r = 10\) into \((x - h)^2+(y - k)^2=r^2\), we get \((x-0)^2+(y - 0)^2=10^2\), which simplifies to \(x^{2}+y^{2}=100\).

Answer:

A. \(x^{2}+y^{2}=100\)