Question

1. this is the equation of a circle. x² + y² - 2x - 4y - 31 = 0 reference hw 2.8, #22

a. complete the square and write the given equation in standard form.
(x - 1)²+(y - 2)² = 36
b. give the center and radius of the circle.
center:
radius: r =
c. graph the circle.

Explanation:

Step1: Recall circle standard - form

The standard - form of a circle equation is $(x - h)^2+(y - k)^2=r^2$, where $(h,k)$ is the center of the circle and $r$ is the radius. The given standard - form is $(x - 1)^2+(y - 2)^2=36$.

Step2: Find the center

Comparing $(x - 1)^2+(y - 2)^2=36$ with $(x - h)^2+(y - k)^2=r^2$, we can see that $h = 1$ and $k = 2$. So the center of the circle is $(1,2)$.

Step3: Find the radius

Since $(x - 1)^2+(y - 2)^2=36$ and $(x - h)^2+(y - k)^2=r^2$, then $r^2=36$. Taking the square - root of both sides (and considering $r>0$), we get $r = 6$.

Answer:

center: $(1,2)$
radius: $r = 6$

To graph the circle:

1. Plot the center point $(1,2)$ on the coordinate plane.
2. From the center, move 6 units in all directions (up, down, left, and right) to mark points on the circle. Then, sketch the circle passing through these points.