Question

this circle is centered at the origin, and the length of its radius is 6. what is the circles equation? a. x^6 + y^6 = 1 b. x + y = 36 c. x^2 + y^2 = 6 d. x^2 + y^2 = 36

Explanation:

Step1: Recall circle - equation formula

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

Step2: Identify center and radius

The circle is centered at the origin \((0,0)\), so \(h = 0\) and \(k = 0\), and the radius \(r = 6\).

Step3: Substitute values into formula

Substituting \(h = 0\), \(k = 0\), and \(r = 6\) into the formula \((x - h)^2+(y - k)^2=r^2\), we get \((x-0)^2+(y - 0)^2=6^2\), which simplifies to \(x^{2}+y^{2}=36\).

Answer:

D. \(x^{2}+y^{2}=36\)