To solve this problem, we are dealing with the formula for the length of the internal tangent between two circles. Given dataTo solve this problem, we are dealing with the formula for the length of the internal tangent between two circles. Given

1. The radius of the larger circle, $r_1 = 9$.
2. The radius of the smaller circle, $r_2 = 6$.
3. The distance between the centers $C$ and $E$, denoted as $d = 10$.

The formula for the length of the internal tangent line (the common internal tangent) between two circles is:

$L = \sqrt{d^2 - (r_1 + r_2)^2}$

Substituting the values:

1. $d = 10$
2. $r_1 + r_2 = 9 + 6 = 15$

Plugging these into the formula:

$L = \sqrt{10^2 - 15^2} = \sqrt{100 - 225} = \sqrt{-125}$

However, this results in a negative square root, which suggests that the internal tangent does not exist under these conditions due to the distance being too small relative to the sum of the radii.

In summary, with $d = 10$, the internal tangent cannot exist because the two circles are too close to each other (or overlapping).

Would you like more details, or do you have any questions?

Here are some additional related questions:

1. How would we calculate the external tangent if $d$ was larger?
2. What is the geometric interpretation if the circles overlap?
3. Can the formula be adjusted if we are looking for an external tangent instead?
4. How would this change if the circles had equal radii?
5. What would happen if $d = r_1 + r_2$, exactly equal to the sum of the radii?

Tip: Always verify the conditions when applying the tangent length formula to check if a solution is feasible.