Question

write the equation of this circle in standard form.

Explanation:

Step1: Identify the center of the circle

The center of the circle is at the point \((h, k)\). From the graph, the center is on the y - axis at \((0, - 6)\). So \(h = 0\) and \(k=-6\).

Step2: Determine the radius of the circle

The radius \(r\) is the distance from the center to a point on the circle. Looking at the graph, the distance from the center \((0, - 6)\) to a point on the circle (e.g., \((0, - 4)\) or \((0, - 8)\)) is \(2\) units. So \(r = 2\).

Step3: Use the standard form of the circle's equation

The standard form of the equation of a circle is \((x - h)^2+(y - k)^2=r^2\). Substituting \(h = 0\), \(k=-6\), and \(r = 2\) into the formula, we get \((x - 0)^2+(y-(-6))^2=2^2\), which simplifies to \(x^2+(y + 6)^2 = 4\).

Answer:

\(x^2+(y + 6)^2=4\)