Question

the endpoints of a diameter of a circle are (-8, -8) and (4, -24). (a) write an equation of the circle in standard form. (b) graph the circle.

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 \((x_1,y_1)=(-8,-8)\) and \((x_2,y_2)=(4,-24)\), then \(h=\frac{-8 + 4}{2}=\frac{-4}{2}=-2\) and \(k=\frac{-8-24}{2}=\frac{-32}{2}=-16\). So the center \((h,k)=(-2,-16)\).

Step2: Find the radius of the circle

The radius \(r\) is the distance from the center \((h,k)\) to either of the endpoints of the diameter. Using the distance formula \(d=\sqrt{(x_2 - x_1)^2+(y_2 - y_1)^2}\), with \((h,k)=(-2,-16)\) and \((x_1,y_1)=(-8,-8)\):
\[

$$\begin{align*} r&=\sqrt{(-2+8)^2+(-16 + 8)^2}\\ &=\sqrt{(6)^2+(-8)^2}\\ &=\sqrt{36 + 64}\\ &=\sqrt{100}\\ &=10 \end{align*}$$

\]

Step3: Write the standard form of the circle equation

The standard form of a circle equation is \((x - h)^2+(y - k)^2=r^2\). Substituting \(h=-2\), \(k=-16\), and \(r = 10\) into the equation, we get \((x+2)^2+(y + 16)^2=100\).

Answer:

(a) \((x + 2)^2+(y + 16)^2 = 100\)
(b) To graph the circle:

1. Identify the center \((h,k)=(-2,-16)\).
2. The radius \(r = 10\). Plot the center point on the coordinate - plane. Then, from the center, move 10 units up, down, left, and right to get four points on the circle. Sketch the circle passing through these points.