Question

write the equation of the circle graphed below.

Explanation:

Step1: Identify the center of the circle

From the graph, the center of the circle \((h,k)\) is at \((2,2)\) (by looking at the coordinates of the red dot).

Step2: Determine the radius of the circle

We can find the radius by calculating the distance from the center \((2,2)\) to a point on the circle, for example, the blue dot at \((5,4)\). Using the distance formula \(r = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), where \((x_1,y_1)=(2,2)\) and \((x_2,y_2)=(5,4)\).

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Step3: Use the standard equation of a circle

The standard equation of a circle is \((x - h)^2+(y - k)^2=r^2\), where \((h,k)\) is the center and \(r\) is the radius. Substituting \(h = 2\), \(k = 2\), and \(r^2=13\) into the equation.

$$ (x - 2)^2+(y - 2)^2 = 13 $$

Answer:

\((x - 2)^2+(y - 2)^2 = 13\)