Question

use the given information to write the equation of the circle in standard form
the endpoints of the diameter are (6,11) and (2,1)

Explanation:

Step1: Find the center of the circle

The center of the circle is the mid - point of the diameter. The mid - point formula is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
Given \((x_1,y_1)=(6,11)\) and \((x_2,y_2)=(2,1)\), then the center \((h,k)=(\frac{6 + 2}{2},\frac{11+1}{2})=(4,6)\)

Step2: Find the radius of the circle

The radius \(r\) is the distance from the center \((h,k)\) to one of the endpoints. The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
Using the center \((4,6)\) and the point \((6,11)\), we have \(r=\sqrt{(6 - 4)^2+(11 - 6)^2}=\sqrt{4 + 25}=\sqrt{29}\)

Step3: Write the equation of the circle in standard form

The standard form of the equation of a circle is \((x - h)^2+(y - k)^2=r^2\)
Substituting \(h = 4\), \(k = 6\) and \(r=\sqrt{29}\) into the formula, we get \((x - 4)^2+(y - 6)^2=29\)

Answer:

\((x - 4)^2+(y - 6)^2=29\)