To solve this problem, we can use the properties of tangent lines to circles. Let's break down the problem into smaller steps:

Step 1: Drawing a diagram
To better understand the problem, let's draw a diagram representing the two circles and the tangent line. Label the larger circle with radius R and the smaller circle with radius r. Draw the tangent line so that it touches both circles externally.

Step 2: Identifying the key elements
From the diagram, we can identify the following key elements:
- Radius of the larger circle: R
- Radius of the smaller circle: r
- Length of the tangent line: l

Step 3: Understanding tangent lines
A tangent line to a circle is perpendicular to the radius at the point of tangency. This means that the tangent line forms a right angle with the radius of the circle at the point where it touches the circle.

Step 4: Using the Pythagorean theorem
Since the tangent line forms a right angle with the radius at the point of tangency, we can use the Pythagorean theorem to relate the lengths of the tangent line, the radius of the larger circle, and the radius of the smaller circle.

In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. In this case, the tangent line is the hypotenuse, and the radii of the circles are the other two sides.

Using the Pythagorean theorem, we can write the equation as:
l^2 = (R + r)^2 - (R - r)^2

Expanding and simplifying the equation, we get:
l^2 = 4Rr

Step 5: Simplifying the equation
To find the relationship between l, R, and r, we need to simplify the equation further.

Taking the square root of both sides, we get:
l = 2√(Rr)

Step 6: Comparing with the given options
We can now compare the simplified equation with the given options:
a) l = R^2 - r^2
b) l = R^2 - r^2
c) l = 2√(Rr)
d) l = 3R^2 - r^2

Option c) matches the simplified equation we derived, so the correct answer is option c) l = 2√(Rr).