T is a point on the common tangents at P of two circles and if TA and TB are respectively the other tangents at A and B to the two circles drawn from the point T then

Options

A. `TA = 2 TB `
B. `TA = TB `
C. `TA = 1/2 TB `
D. `3TA = TB `

Answer

To solve the problem step by step, we will analyze the given information about the two circles and the tangents drawn from point T.

### Step-by-Step Solution:

1. **Identify the Circles and Tangents**:
- Let Circle 1 and Circle 2 be the two circles.
- Point T lies on the common tangent at point P, which touches both circles.

2. **Draw the Tangents**:
- From point T, draw the tangent TA to Circle 1 at point A.
- Similarly, draw the tangent TB to Circle 2 at point B.

3. **Use the Tangent Property**:
- According to the property of tangents from a point outside a circle, the lengths of the tangents drawn from a single external point to a circle are equal.
- Therefore, for Circle 1, we have:
\[ TA = TP \]
- For Circle 2, we have:
\[ TB = TP \]

4. **Set Up the Equations**:
- From the first circle, we have:
\[ TA = TP \quad \text{(Equation 1)} \]
- From the second circle, we have:
\[ TB = TP \quad \text{(Equation 2)} \]

5. **Equate the Tangents**:
- Since both TA and TB are equal to TP, we can equate them:
\[ TA = TB \]

6. **Conclusion**:
- Therefore, the lengths of the tangents from point T to both circles are equal:
\[ TA = TB \]

### Final Answer:
The correct conclusion is that \( TA = TB \).