Question

what is the standard form of the equation for the circle?
a. $(x - 3)^2+(y - 2)^2 = 8$
b. $(x + 3)^2+(y + 2)^2 = 8$
c. $(x - 3)^2+(y - 2)^2 = 32$
d. $(x + 3)^2+(y + 2)^2 = 32$

Explanation:

Step1: Recall the standard form of a circle equation

The standard form of a circle 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

From the graph, the center of the circle \(A=(h,k)=(3,2)\).

Step3: Calculate the radius

Use the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\) to find the radius. Let \(A(3,2)\) be \((x_1,y_1)\) and \(B(5,4)\) be \((x_2,y_2)\). Then \(r=\sqrt{(5 - 3)^2+(4 - 2)^2}=\sqrt{4 + 4}=\sqrt{8}\). So \(r^2 = 8\).

Step4: Substitute into the standard form

Substitute \(h = 3\), \(k = 2\), and \(r^2=8\) into \((x - h)^2+(y - k)^2=r^2\), we get \((x - 3)^2+(y - 2)^2=8\).

Answer:

A. \((x - 3)^2+(y - 2)^2=8\)