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 the center

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 \(x = 7\) (and \(x = -7\)) and the \(y\)-axis at \(y = 7\) (and \(y = -7\)), so the radius \(r = 7\).

Step4: Substitute into the formula

Substitute \(h = 0\), \(k = 0\), and \(r = 7\) into \((x - h)^2 + (y - k)^2 = r^2\). We get \((x - 0)^2 + (y - 0)^2 = 7^2\), which simplifies to \(x^2 + y^2 = 49\).

Answer:

\(x^2 + y^2 = 49\)