Question

use the given information about a circle to write an equation of the circle in standard form.
the endpoints of a diameter are (3, 11) and (-9, -5).
a) $(x - 3)^2 + (y + 3)^2 = 20$
b) $(x - 3)^2 + (y - 11)^2 = 100$
c) $(x + 3)^2 + (y - 3)^2 = 100$
d) $(x + 3)^2 + (y + 11)^2 = 20$

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 \((3,11)\) and \((-9,-5)\), we have \(x=\frac{3+( - 9)}{2}=\frac{-6}{2}=-3\), \(y=\frac{11+( - 5)}{2}=\frac{6}{2}=3\). So the center \((h,k)=(-3,3)\).

Step2: Find the radius of the circle

The radius \(r\) is half of the length of the diameter.
The distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For the points \((3,11)\) and \((-9,-5)\), \(d=\sqrt{(-9 - 3)^2+(-5 - 11)^2}=\sqrt{(-12)^2+(-16)^2}=\sqrt{144 + 256}=\sqrt{400}=20\).
So \(r=\frac{d}{2}=10\).

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 \(h=-3,k = 3,r = 10\) into the formula, we get \((x+3)^2+(y - 3)^2=100\).

Answer:

C. \((x + 3)^2+(y - 3)^2=100\)