Question

Find an equation of the line containing the centers of the two circles

x2 + y2 − 4x + 6y + 4 = 0 and

x2 + y2 + 6x + 4y + 9 = 0
I have an answer for this 1 but I want to make sure I am right.

Answer 1

What are the points of the centers of the circles you have?

Answer 2

Do you know how to find the centres of the circles?
Circle one is (2,-3) I believe

Answer 3

The ? I asked is the only thing on the paper so I dont know what the points are

Answer 4

Oh, you said you had an answer. Would you like me to show you how to find the centres of the circles?

Answer 5

Yes I would, I came up with Neither

Answer 6

You have to 'complete the square' for both the x terms and y terms. This is because you need the equation in standard form.
\[x ^{2}+y ^{2}-4x+6y+4=0\]
Gather
\[x ^{2}-4x+y ^{2}+6y+4=0\]
\[(x ^{2}-4x)+(y ^{2}+6y)+4=0\]
Now comes the complete the square part
\[(x ^{2}-4x+4-4)+(y ^{2}+6y+9-9)+4=0\]
\[(x ^{2}-4x+4)-4+(y ^{2}+6y+9)-9+4=0\]
simplify
\[(x ^{2}-4x+4)+(y ^{2}+6y+9)-9=0\]
Factor our x and y terms
\[(x-2)^{2}+(y+3)^{2}=9\]
this means the centre of the circle is (2,-3)