Question

write the equation of this circle in standard form.

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 coordinates

From the graph, the center of the circle is at the origin \((0, 0)\), so \(h = 0\) and \(k = 0\).

Step3: Determine the radius

The circle intersects the \(x\)-axis at \(x = 4\) and \(x = -4\), so the radius \(r\) is the distance from the center \((0,0)\) to \((4,0)\), which is \(r = 4\).

Step4: Substitute into standard form

Substitute \(h = 0\), \(k = 0\), and \(r = 4\) into \((x - h)^2 + (y - k)^2 = r^2\). We get \((x - 0)^2 + (y - 0)^2 = 4^2\), which simplifies to \(x^2 + y^2 = 16\).

Answer:

\(x^2 + y^2 = 16\)