Question

give the equation of a circle with a diameter that has endpoints (2, 3) and (8, -8).

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 \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
For the points \((x_1,y_1)=(2,3)\) and \((x_2,y_2)=(8,-8)\), we have \(x=\frac{2 + 8}{2}=\frac{10}{2}=5\) and \(y=\frac{3+( - 8)}{2}=\frac{3 - 8}{2}=\frac{-5}{2}=-2.5\). So the center \((h,k)=(5,-2.5)\).

Step2: Find the radius of the circle

The radius \(r\) is half of the length of the diameter. The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
The length of the diameter \(d=\sqrt{(8 - 2)^2+(-8 - 3)^2}=\sqrt{6^2+(-11)^2}=\sqrt{36 + 121}=\sqrt{157}\). So the radius \(r=\frac{\sqrt{157}}{2}\).

Step3: Write the equation of the circle

The standard form of the equation of a circle is \((x - h)^2+(y - k)^2=r^2\).
Substituting \(h = 5\), \(k=-2.5\) and \(r=\frac{\sqrt{157}}{2}\) into the formula, we get \((x - 5)^2+(y + 2.5)^2=\frac{157}{4}\).

Answer:

\((x - 5)^2+(y+\frac{5}{2})^2=\frac{157}{4}\)