Given:
Circle equation: x^2 + y^2 - 4x - 6y - 12 = 0

To Find:
The coordinates of the point where x = 7 touches the circle.

Solution:

Step 1: Rewrite the Circle Equation
We can rewrite the given circle equation in standard form by completing the square for both x and y terms.

x^2 - 4x + y^2 - 6y = 12

Step 2: Completing the Square for x
To complete the square for the x terms, we need to add and subtract the square of half the coefficient of x (which is -4/2 = -2) inside the parentheses.

x^2 - 4x + 4 + y^2 - 6y = 12 + 4

Simplifying the equation:

(x - 2)^2 + y^2 - 6y = 16

Step 3: Completing the Square for y
To complete the square for the y terms, we need to add and subtract the square of half the coefficient of y (which is -6/2 = -3) inside the parentheses.

(x - 2)^2 + y^2 - 6y + 9 = 16 + 9

Simplifying the equation:

(x - 2)^2 + (y - 3)^2 = 25

The equation is now in the standard form of a circle:

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

Where the center of the circle is at point (h, k) and the radius is r.

Step 4: Determine the Center and Radius
Comparing the given equation with the standard form, we can identify the center and radius as follows:

Center: (h, k) = (2, 3)
Radius: r = sqrt(25) = 5

Step 5: Find the Point of Contact
Since we are given that x = 7 touches the circle, the y-coordinate of the point of contact can be found by substituting x = 7 into the equation of the circle:

(7 - 2)^2 + (y - 3)^2 = 25

Simplifying the equation:

25 + (y - 3)^2 = 25

(y - 3)^2 = 0

Taking the square root of both sides:

y - 3 = 0

y = 3

Therefore, the point of contact is (7, 3).

Conclusion:
The coordinates of the point where x = 7 touches the circle x^2 + y^2 - 4x - 6y - 12 = 0 are (7, 3).