Question

determine the equation of the circle graphed below.

Explanation:

Step1: Recall the standard circle equation

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 of the circle

From the graph, the center of the circle appears to be at \((4, 2)\) (by visually inspecting the midpoint of the circle's symmetry).

Step3: Calculate the radius

To find the radius, we can use the distance from the center \((4, 2)\) to a point on the circle, say \((9, 2)\) (since it's on the x - axis and easy to calculate distance). The distance formula between \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\). For \((4,2)\) and \((9,2)\), \(d=\sqrt{(9 - 4)^2+(2 - 2)^2}=\sqrt{5^2+0^2} = 5\). So the radius \(r = 5\).

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

Substitute \(h = 4\), \(k = 2\), and \(r = 5\) into \((x - h)^2+(y - k)^2=r^2\). We get \((x - 4)^2+(y - 2)^2=5^2=25\).

Answer:

\((x - 4)^2+(y - 2)^2 = 25\)