To determine the relationship between the two circles, let's solve the given equations step by step.

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

First, let's rewrite Equation 1 in standard form by completing the square for x:

x^2 - 6x + y^2 = 0
(x^2 - 6x + 9) + y^2 = 9 (Adding 9 on both sides to complete the square)
(x - 3)^2 + y^2 = 3^2

Similarly, let's rewrite Equation 2:

x^2 - 2x + y^2 = 0
(x^2 - 2x + 1) + y^2 = 1 (Adding 1 on both sides to complete the square)
(x - 1)^2 + y^2 = 1^2

Now we can compare the equations with the standard form of a circle:

Equation 1: (x - h)^2 + (y - k)^2 = r^2
Equation 2: (x - a)^2 + (y - b)^2 = R^2

From Equation 1, we can identify the center of the first circle as (h, k) = (3, 0) and the radius as r = 3.

From Equation 2, we can identify the center of the second circle as (a, b) = (1, 0) and the radius as R = 1.

Now let's analyze the relationship between the two circles based on their centers and radii.

1. Centers of the circles:
The centers of the two circles are different, (3, 0) and (1, 0), indicating that the circles are not concentric.

2. Distance between the centers:
The distance between the centers of the circles is d = √[(a - h)^2 + (b - k)^2] = √[(1 - 3)^2 + (0 - 0)^2] = √4 = 2.

3. Relationship between the radii:
The difference between the radii is R - r = 1 - 3 = -2, which is less than the sum of the radii, indicating that the circles touch each other externally.

Therefore, the correct answer is option A - They touch each other externally.

Now, let's discuss the area of the triangle formed by their common tangents.

To find the common tangents, we need to draw the lines tangent to both circles at their points of contact.

The points of contact are the points where the circles touch each other externally. In this case, it's the single point (3, 0).

The common tangents are the lines passing through the point of contact and perpendicular to the line joining the centers of the circles.

Since the centers of the circles are (3, 0) and (1, 0), the line joining the centers is y = 0.

The equations of the common tangents passing through (3, 0) and perpendicular to y = 0 are x = 3 and x = 1.

To find the area of the triangle formed by these common tangents, we need to find the