Two circles have a common tangent touching them at point P. From an external point T on this common tangent, tangents TA and TP are drawn to the first circle and tangents TB and TP are drawn to the second circle. If TA and TB are the other tangents from T to the two circles, what is the relationship between TA and TB?

Introduction / Context:
This is a pure geometry question about tangent segments drawn from an external point to a circle. It uses a fundamental property that the lengths of two tangent segments drawn from the same external point to a circle are equal. Here we have two circles and a common external point, and the problem asks about the equality of tangent lengths from that point to each circle.

Given Data / Assumptions:

• Two circles are given, with a common tangent touching them at a point P.
• T is a point on this common tangent.
• From T, tangents to the first circle touch it at P and A, giving segments TP and TA.
• From T, tangents to the second circle touch it at P and B, giving segments TP and TB.
• TA and TB are the second tangent segments from T to the respective circles.

Concept / Approach:
The key property is: from a fixed external point to a given circle, all tangent segments drawn to that circle have equal length. Therefore, for the first circle, TA and TP are equal. For the second circle, TB and TP are equal. Since both TA and TB are equal in length to TP, they must be equal to each other. This conclusion holds regardless of the sizes of the two circles as long as the tangents are correctly drawn from the same external point T.

Step-by-Step Solution:
Step 1: For the first circle, note that T is an external point and TA and TP are tangents drawn from T to that circle. Step 2: By the tangent property, TA = TP for the first circle. Step 3: For the second circle, T is again an external point and TB and TP are tangents drawn from T to the second circle. Step 4: By the same property, TB = TP for the second circle. Step 5: Since TA = TP and TB = TP, it follows that TA = TB.

Verification / Alternative check:
If we imagine specific circle sizes and draw them accurately, measuring the tangent segments from T to each circle would reveal that the distance from T to each point of tangency on the same circle is equal. This is a direct consequence of congruent right triangles formed between the centre, the point of tangency and the external point. The two circles are independent in size, but the equality for each circle separately and the common value TP force TA and TB to be equal.

Why Other Options Are Wrong:
Options stating TA = 2 TB, TA = 1/2 TB or 3TA = TB contradict the fundamental tangents from a point property. There is no reason for a constant ratio other than 1 between TA and TB, since each individually equals TP. The statement that TA + TB is constant but TA not equal to TB has no geometric basis here. Only the equality TA = TB matches the precise theorem about tangent segments from a fixed external point.

Common Pitfalls:
Learners may get distracted by the presence of two different circles and think that tangent lengths must somehow depend on radii. In reality, equality is always between tangents from the same external point to the same circle. Another mistake is to confuse the common tangent at P with the extra tangents TA and TB. Keeping the diagram clear and remembering the core property of tangent segments helps avoid such confusion.

Final Answer:
The correct relationship is TA = TB.