Question

Part 2: Identify the center and radius of each. Then sketch the graph.

1. x^2 + y^2 - 4x + 6y + 9 = 0

2. x^2 + y^2 + 6x + 2y + 9 = 0

3. x^2 + y^2 - 6x + 2y + 1 = 0

4. x^2 + y^2 - 8x -2y + 8 = 0

Answer 1

See below for graphs

1. (2, -3), r = 2 (blue)
2. (-3, -1), r = 1 (green)
3. (3, -1), r = 3 (purple)
4. (4, 1), r = 3 (red)

Explanation:

I like to solve a set of problems like this once, then use a calculator or spreadsheet to crunch the numbers.

We can start with the general form of the equation for a circle:

x^2 +y^2 +ax +by +c = 0

Completing the square gives ...

(x^2 +ax +(a/2)^2) +(y^2 +by +(b/2)^2) = (a/2)^2 +(b/2)^2 -c

(x +(a/2))^2 +(y +(b/2))^2 = (a/2)^2 +(b/2)^2 -c

Comparing this to the standard form formula ...

(x -h)^2 + (y -k)^2 = r^2

we see the relations are ...

• h = -a/2
• k = -b/2
• r = √((a/2)² +(b/2)² -c)

where (h, k) is the center and r is the radius.

Then your circle centers and radii are ...

1. (h, k) = (-(-4)/2, -6/2) = (2, -3); r = √(4+9-9) = 2
2. (h, k) = (-6/2, -2/2) = (-3, -1); r = √(9+1-9) = 1
3. (h, k) = (-(-6)/2, -2/2) = (3, -1); r = √(9+1-1) = 3
4. (h, k) = (-(-8)/2, -(-2)/2) = (4, 1); r = √(16+1-8) = 3