Question

use the graph below to determine the equation of the circle in (a) center - radius form and (b) general form.
a. type the equation in center - radius form.
(simplify your answer.)

Explanation:

Step1: Find the center of the circle

The center of the circle is \((h,k)=(1, - 3)\) (since the center is the mid - point of the diameter endpoints. For example, if we consider two points \((x_1,y_1)=(3,1)\) and \((x_2,y_2)=(3, - 5)\), the \(x\) - coordinate of the center \(h=\frac{3 + 3}{2}=1\) (using the mid - point formula \(h=\frac{x_1+x_2}{2}\), \(k=\frac{y_1 + y_2}{2}\), and also verified by symmetry in the graph).

Step2: Find the radius of the circle

The radius \(r\) is the distance from the center \((h,k)=(1,-3)\) to a point on the circle, say \((3,1)\). Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), we have \(r=\sqrt{(3 - 1)^2+(1+3)^2}=\sqrt{4 + 16}=\sqrt{20}=2\sqrt{5}\)

Step3: Write the equation in center - radius form

The center - radius form of a circle is \((x - h)^2+(y - k)^2=r^2\). Substituting \(h = 1\), \(k=-3\) and \(r = 2\sqrt{5}\), we get \((x - 1)^2+(y + 3)^2=20\)

Answer:

\((x - 1)^2+(y + 3)^2=20\)