Question

find the standard form for the equation of a circle $(x - h)^2+(y - k)^2=r^2$ with a diameter that has endpoints $(-3,1)$ and $(1,-5)$. $h =$ $k =$ $r=$ question help: video 1 video 2

Explanation:

Step1: Find the center of the circle (h,k)

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 endpoints \((x_1=-3,y_1 = 1)\) and \((x_2 = 1,y_2=-5)\), we have:
\(h=\frac{-3 + 1}{2}=\frac{-2}{2}=-1\)
\(k=\frac{1+( - 5)}{2}=\frac{1 - 5}{2}=\frac{-4}{2}=-2\)

Step2: Find the radius r

The radius is half the length of the diameter. The distance formula between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
First, find the length of the diameter:
\(d=\sqrt{(1-( - 3))^2+(-5 - 1)^2}=\sqrt{(1 + 3)^2+(-6)^2}=\sqrt{16 + 36}=\sqrt{52}=2\sqrt{13}\)
Then \(r=\frac{d}{2}=\sqrt{13}\)

Answer:

\(h=-1\), \(k=-2\), \(r = \sqrt{13}\)