x² + y² + 4x - 2y = 4
Rearrange the terms and skip some spaces:
x² + 4x + y² - 2y = 4
Take half the coefficient of x which is 4, get 2
Square 2, get 4. Add + 4 in the first space on
the left and also add + 4 to the right side:
x² + 4x + 4 + y² - 2y = 4 + 4
Take half the coefficient of y which is -2, get -1
Square -1, get +1. Add + 1 in the second space on
the left and also add + 1 to the right side:
x² + 4x + 4 + y² - 2y + 1 = 4 + 4 + 1
Combine the terms on the right side
x² + 4x + 4 + y² - 2y + 1 = 9
Factor the first three terms on the left as
(x + 2)(x + 2) or (x + 2)²
(x + 2)² + y² - 2y + 1 = 9
Factor the last three terms on the left as
(y - 1)(y - 1) or (y - 1)²
(x + 2)² + (y - 1)² = 9
Compare that to
(x - h)² + (y - k)² = r²
-h = +2 so h = -2
-k = -1 so k = 1
r² = 9 so r = 3
So this is a circle with center (h,k) = (-2,1)
and radius 3
So we get a compass and stick the sharp point
on the green point below:
Then open out the compass to a radius of 3 and
swing it around to draw this circle:
To check it we can see that the graph goes
through these four points:
(-5,1), (-2,4), (1,1) and (-2,-2)
We check those points to see if they all
satisfy the original equation:
x² + y² + 4x - 2y = 4
Checking (-5,1)
5² + (-1)² + 4(-5) - 2(1) = 4
25 + 1 - 20 - 2 = 4
26 - 20 - 2 = 4
6 - 2 = 4
4 = 4
That one does.
Checking (-2,4)
(-2)² + 4² + 4(-2) - 2(4) = 4
4 + 16 - 8 - 8 = 4
20 - 8 - 8 = 4
12 - 8 = 4
4 = 4
That one does.
Checking (1,1)
1² + 1² + 4(1) - 2(1) = 4
1 + 1 + 4 - 2 = 4
2 + 4 - 2 = 4
6 - 2 = 4
4 = 4
That one does.
Checking (-2,-2)
(-2)² + (-2)² + 4(-2) - 2(-2) = 4
4 + 4 - 8 + 4 = 4
8 - 8 + 4 = 4
4 = 4
That one does too. So now we know we have the
right circle.
Edwin
