Multiple choice

lf $S_{1}=0,\ S_{2}=0$ and $S_{3}=0$ are the three circles whose radical centre is the point $\mathrm{P}$, then the lengths $l_{1},\ l_{2},\ l_{3}$ of the tangent from $\mathrm{P}$ to the three circles are such that

  1. $l_{1}=21_{2}=3l_{3}$
  2. $l_{1}=l_{2}=l_{3}$
  3. $l_{1}\neq l_{2}\neq l_{3}$
  4. $l_{1}=l_{2}\neq l_{3}$
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B Correct answer
Explanation

At the radical centre, the powers with respect to all three circles are equal. Tangent length squared equals the corresponding power, so the three tangent lengths are equal.

AI explanation

The radical centre of three circles is defined as the unique point from which the tangent lengths drawn to all three circles are equal. Let the lengths of the tangents from point P to the three circles S1=0, S2=0 and S3=0 be l1, l2 and l3 respectively. By the fundamental property of the radical centre, these lengths must be identical, meaning l1 = l2 = l3.

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