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The lengths of tangents from a fixed poi...

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`.

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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.
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P, Q, R are the centres and r_(1),r_(2),r_(3) are the radii respectively of three coaxial circles, then QR r_(1)^(2)+RP r_(2)^(2) +PQ r_(3)^(2)=PQ.QR.RP

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Knowledge Check

  • Statement :1 If a parabola y ^(2) = 4ax intersects a circle in three co-normal points then the circle also passes through the vertr of the parabola. Because Statement : 2 If the parabola intersects circle in four points t _(1), t_(2), t_(3) and t_(4) then t _(1) + t_(2) + t_(3) +t_(4) =0 and for co-normal points t _(1), t_(2) , t_(3) we have t_(1)+t_(2) +t_(3)=0.

    A
    Stateme-1 is True, Statement-2 is True, Statemetn-2 is correct explanation for Statement-1
    B
    Statement-1 is True, Statement-2 is True, Statement-2 is NOT a correct explanation for Statement-1
    C
    Statement-1 is True, Statement-2 is False
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