Question

sketch the graph of the following circle.$(x - 4)^2 + (y - 5)^2 = 64$(a) find the center of the circle.(b) find the radius of the circle.(c) graph the circle.(a) find the center of the circle.(type an ordered pair.)

Explanation:

Step1: Match to circle standard form

The standard form of a circle is $(x-h)^2+(y-k)^2=r^2$, where $(h,k)$ is the center and $r$ is the radius. Compare to $(x-4)^2+(y-5)^2=64$.

Step2: Identify center coordinates

From the comparison, $h=4$ and $k=5$, so the center is $(4,5)$.

Step3: Calculate the radius

Since $r^2=64$, solve for $r$: $r=\sqrt{64}=8$.

Step4: Describe graphing steps

Plot the center $(4,5)$, then mark points 8 units up, down, left, and right from the center: $(4,13)$, $(4,-3)$, $(-4,5)$, $(12,5)$. Draw a smooth circle through these points.

Answer:

(a) $(4,5)$
(b) $8$
(c) To graph:

1. Plot the center point $(4,5)$ on the coordinate plane.
2. Plot four key points: 8 units above $(4,13)$, 8 units below $(4,-3)$, 8 units left $(-4,5)$, and 8 units right $(12,5)$ from the center.
3. Draw a smooth, closed circle passing through all four key points.