Question

what is the equation of the circle shown in the graph? (x - 3)^2 + y^2 = 4 (x + 3)^2 + y^2 = 2 x^2 + (y + 3)^2 = 4 x^2 + (y - 3)^2 = 2

Explanation:

Step1: Recall the standard equation of a circle

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

Step2: Identify the center and radius from the graph

From the graph, the center of the circle is \((0,- 3)\) (i.e., \(h = 0,k=-3\)) and the radius \(r = 2\) (since the distance from the center to a point on the circle is \(2\) units).

Step3: Substitute \(h,k,r\) into the standard equation

Substitute \(h = 0,k=-3,r = 2\) into \((x - h)^2+(y - k)^2=r^2\).
We get \((x - 0)^2+(y-(-3))^2=2^2\), which simplifies to \(x^{2}+(y + 3)^{2}=4\).

Answer:

\(x^{2}+(y + 3)^{2}=4\) (the third option)