The lengths of tangents from a fixed point to three circles of coaxial system are `t_(1),t_(2),t_(3)` and if P, Q, R be the centres, then `QR t_(1)^(2) +RP t_(2)^(2) +PQ t_(3)^(2)=0`.

Answer

To solve the problem, we need to analyze the given equation involving the lengths of tangents from a fixed point to three circles of a coaxial system. The equation is:

\[ QR \cdot t_1^2 + RP \cdot t_2^2 + PQ \cdot t_3^2 = 0 \]

### Step-by-Step Solution:

1. **Understanding the Terms**:
- Let \( t_1, t_2, t_3 \) be the lengths of tangents from a fixed point to the three circles.
- Let \( P, Q, R \) be the centers of the circles corresponding to \( t_1, t_2, t_3 \) respectively.
- The terms \( QR, RP, PQ \) represent the distances between the centers of the circles.

2. **Setting Up the Equation**:
- The equation states that the weighted sum of the squares of the tangent lengths is equal to zero. This implies a relationship between the distances between the centers and the lengths of the tangents.

3. **Analyzing the Equation**:
- Since \( t_1, t_2, t_3 \) are lengths, they are non-negative. Therefore, for the equation to hold true, the coefficients \( QR, RP, PQ \) must balance the terms such that their weighted sum results in zero.

4. **Interpreting the Result**:
- The equation can be interpreted as a condition for the coaxial circles. If the circles are coaxial, the lengths of the tangents from any external point must satisfy this relationship.
- This means that if one of the tangent lengths is increased or decreased, the others must adjust accordingly to maintain the equality.

5. **Conclusion**:
- The equation \( QR \cdot t_1^2 + RP \cdot t_2^2 + PQ \cdot t_3^2 = 0 \) indicates a specific geometric relationship among the circles and the point from which the tangents are drawn. It reflects the balance of distances and tangent lengths in a coaxial system.