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 \((h, k)\)

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

Step3: Determine radius \(r\)

The circle extends from \(x = -2\) to \(x = 2\) (or \(y = -2\) to \(y = 2\)), so the radius \(r = 2\) (distance from center \((0,0)\) to a point on the circle, e.g., \((2, 0)\)).

Step4: Substitute \(h\), \(k\), and \(r\) into the formula

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

Answer:

\(x^2 + y^2 = 4\)