Question

for a circle defined by the given equation, what are the coordinates of the center and the length of the radius?
$x^2 + y^2 - 4x - 10y + 20 = 0$
a. center: (-2,-5)
radius: 9 units
b. center: (2,5)
radius: 3 units
c. center: (2,5)
radius: 9 units
d. center: (-2,-5)
radius: 3 units

Explanation:

Step1: Rearrange the equation

$x^2 - 4x + y^2 - 10y = -20$

Step2: Complete the square for $x$

$x^2 - 4x + 4 = (x-2)^2$, add 4 to right side
$y^2 - 10y + 25 = (y-5)^2$, add 25 to right side

Step3: Rewrite in circle standard form

$(x-2)^2 + (y-5)^2 = -20 + 4 + 25$
$(x-2)^2 + (y-5)^2 = 9$

Step4: Identify center and radius

Circle standard form: $(x-h)^2+(y-k)^2=r^2$, where $(h,k)$ is center, $r$ is radius.
So $h=2, k=5$, $r=\sqrt{9}=3$

Answer:

B. center: (2,5)
radius: 3 units