Question

find the center and radius of the circle. write the standard form of the equation.
the center of the circle is (h,k) = \square.
(type an ordered pair.)
the radius of the circle is r = \square.
the equation of the circle in standard form is \square.
(type your answer in standard form.)

Explanation:

Step1: Identify the center

From the graph, the center of the circle is marked as \((-2, 3)\). So, \((h, k)=(-2, 3)\).

Step2: Calculate the radius

The radius is the distance between the center \((-2, 3)\) and a point on the circle, say \((-2, 0)\). Using the distance formula \(r = \sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), here \(x_1=-2,y_1 = 3,x_2=-2,y_2 = 0\). So \(r=\sqrt{(-2 - (-2))^2+(0 - 3)^2}=\sqrt{0 + 9}=3\).

Step3: Write the standard form

The standard form of a circle's equation is \((x - h)^2+(y - k)^2=r^2\). Substituting \(h=-2\), \(k = 3\) and \(r = 3\), we get \((x+2)^2+(y - 3)^2=9\).

Answer:

The center of the circle is \((h,k)=\boldsymbol{(-2, 3)}\).
The radius of the circle is \(r=\boldsymbol{3}\).
The equation of the circle in standard form is \(\boldsymbol{(x + 2)^2+(y - 3)^2=9}\).