Question

11.1: distance and midpoint formulas; c
score: 5/24 answered: 5/18
question 6
find the standard form for the equation of a circle
$(x - h)^2 + (y - k)^2 = r^2$
with a diameter that has endpoints $(-8, 10)$ and $(2, 6)$.
$h = $
$k = $
$r = $

Explanation:

Step1: Find the center \((h,k)\)

The center of the circle is the midpoint of the diameter. The midpoint formula is \((\frac{x_1 + x_2}{2},\frac{y_1 + y_2}{2})\).
For the points \((-8,10)\) and \((2,6)\), \(h=\frac{-8 + 2}{2}=\frac{-6}{2}=-3\), \(k=\frac{10+6}{2}=\frac{16}{2} = 8\).

Step2: Find the radius \(r\)

The radius is half the length of the diameter. The distance formula is \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\).
For the points \((-8,10)\) and \((2,6)\), \(d=\sqrt{(2-(-8))^2+(6 - 10)^2}=\sqrt{(10)^2+(-4)^2}=\sqrt{100 + 16}=\sqrt{116}=2\sqrt{29}\).
So \(r=\frac{d}{2}=\sqrt{29}\).

Answer:

\(h=-3\), \(k = 8\), \(r=\sqrt{29}\)