Find the equation of the line that contains the centers of t

Find the equation of the line that contains the centers of two circles:

x^2 + y^2 - 4x + 6y + 4 = 0

x^2 + y^2 + 6x + 4y + 9 = 0

Write the line in slope-intercept form.

HINT:

i. Find the centers for each circle.

ii. Find the slope for the line passing through the points.

iii. Find the equation of line containing the points.

x^2 + y^2 - 4x + 6y + 4 = 0 ----(1)

x^2 + y^2 + 6x + 4y + 9 = 0 -----(2)

i) Equation 1: x^2 + y^2 - 4x + 6y + 4 = 0

(x^2 -4x) + (y^2 +6y) +4 =0

(x^2 -4x +4 -4) + ( y^2 +6y + 9 -9) +4 =0

(x-2)^2 + ( y +3)^2 =9

Centre ( 2, -3) and radius = 3 units

Equation 2: x^2 + y^2 + 6x + 4y + 9 = 0

Rearranging : x^2 +6x +9 -9 +y^2 +4y +4 -4 +9 =0

(x +3)^2 + ( y+2)^2 =4

Centre : ( -3, -2) Radius : = 2units

ii) ( 2, -3) and ( -3, -2)

slope: ( -2 +3)/( -3 -2) = -1/5

iii) Line passes through the centre points.So we can use of these points to solve the equation of line:

y = mx+ c ( 2, -3)

y = -x/5 + c ( 2, -3)

-3 = -2/5 +c---> c = -3 +2/5 = -13/5

y = -x/5 -13/5

5y = -x -13

5y +x+13 =0