Solution:

Given,
x2 + y2 - 12x -12y = 0 …(1)
x2 + y2 + 6x + 6y = 0 …(2)

Let us convert the given equations into standard form by completing the square method.

Completing the square method:
Given equation is of the form ax2 + by2 + 2gx + 2fy + c = 0

Step 1: Move the constant term ‘c’ to the right-hand side of the equation
ax2 + by2 + 2gx + 2fy = -c

Step 2: Divide the entire equation by the coefficient of the x2 term, ‘a’
x2 + (b/a)y2 + (2g/a)x + (2f/a)y = -c/a

Step 3: Complete the square for the x terms by adding (g/a)2 to both sides of the equation.
x2 + (b/a)y2 + (2g/a)x + (g/a)2 + (2f/a)y = (g/a)2 - c/a

Step 4: Factor the left-hand side of the equation.
(x + g/a)2 + (b/a)y2 + (2f/a)y = (g/a)2 - c/a

Step 5: Complete the square for the y terms by adding (f/a)2 to both sides of the equation.
(x + g/a)2 + (b/a)(y + f/b)2 = (g/a)2 - c/a + (f/b)2

Now, we will convert equations (1) and (2) into standard form.

Equation (1):
x2 + y2 - 12x -12y = 0
x2 - 12x + y2 - 12y = 0
(x - 6)2 - 36 + (y - 6)2 - 36 = 0
(x - 6)2 + (y - 6)2 = 72

Equation (2):
x2 + y2 + 6x + 6y = 0
x2 + 6x + y2 + 6y = 0
(x + 3)2 - 9 + (y + 3)2 - 9 = 0
(x + 3)2 + (y + 3)2 = 18

Now, we will plot the graphs of the given equations and analyze them.

The graph of equation (1) is a circle with center (6, 6) and radius √72, as (x - 6)2 + (y - 6)2 = 72.

The graph of equation (2) is a circle with center (-3, -3) and radius √18, as (x + 3)2 + (y + 3)2 = 18.

The circles touch each other externally if the distance between their centers is equal to the sum of their radii.

Distance between the centers = √[(6 - (-3))2 + (6 - (-3))2] = √(81 + 81) = √162

Sum of the radii = √72 + √18

Since √162 = √72 + √

If C1C2=R1+R2 ,then 2 circles touch each other externally.