Two circles Intersect at A and B. P is a point on produced BA. PT and PQ are tangents to the circles. The relation of PT and PQ is

Options

A. `PT = 2PQ `
B. `PT lt PQ `
C. `PT gt PQ`
D. `PT = PQ `

Answer

To solve the problem, we need to analyze the relationship between the tangents PT and PQ drawn from point P to the two circles that intersect at points A and B.

### Step-by-Step Solution:

1. **Identify the Elements**:
- Let the two circles intersect at points A and B.
- Point P is on the line extended from segment AB.
- PT is a tangent to one circle from point P.
- PQ is a tangent to the other circle from point P.

2. **Use the Tangent-Secant Theorem**:
- According to the tangent-secant theorem, if a tangent is drawn from a point outside a circle, the square of the length of the tangent segment is equal to the product of the lengths of the whole secant segment and its external part.
- For the first circle, we have:
\[ PT^2 = PA \cdot PB \]
- For the second circle, we have:
\[ PQ^2 = PA \cdot PB \]

3. **Set the Equations Equal**:
- Since both PT and PQ are tangents from point P to their respective circles, we can equate the two equations:
\[ PT^2 = PQ^2 \]

4. **Conclude the Relationship**:
- Taking the square root of both sides gives us:
\[ PT = PQ \]
- This means that the lengths of the tangents PT and PQ are equal.

### Final Answer:
The relation between the tangents PT and PQ is:
\[ PT = PQ \]