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: Recall circle standard form

The standard equation of a circle is $(x-h)^2+(y-k)^2=r^2$, where $(h,k)$ is the center, and $r$ is the radius.

Step2: Identify center (a)

Compare $(x-4)^2+(y-5)^2=64$ to the standard form: $h=4$, $k=5$. So center is $(4,5)$.

Step3: Calculate radius (b)

From the equation, $r^2=64$. Solve for $r$:
$$r=\sqrt{64}=8$$

Step4: Describe graph (c)

Plot the center $(4,5)$ first. 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 passing through these points.

Answer:

(a) $(4, 5)$
(b) $8$
(c) A circle with center $(4, 5)$ and radius 8, plotted on a coordinate plane.