Question

write the equation in standard form for the circle passing through (0, 8) centered at the origin.

Explanation:

Step1: Recall circle standard form

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

Step2: Identify center and radius

The center is at the origin \((0, 0)\), so \(h = 0\), \(k = 0\). The circle passes through \((0, 8)\), so the radius \(r\) is the distance from \((0, 0)\) to \((0, 8)\), which is \(8\). Thus, \(r^2 = 8^2 = 64\).

Step3: Substitute into standard form

Substituting \(h = 0\), \(k = 0\), and \(r^2 = 64\) into the standard form, we get \((x - 0)^2 + (y - 0)^2 = 64\), which simplifies to \(x^2 + y^2 = 64\).

Answer:

\(x^2 + y^2 = 64\)