Multiple choice

Let $C$ be a circle with center '$ O$' and $HK$ is the chord of contact of pair of the tangents from point $A$. $OA$ intersects the circles $C$ at $P$ and $Q$ and $B$ is the midpoint of $HK$, then Statement - 1: $AB$ is the harmonic mean $AP$ and $AQ$ Because Statement - 2: $AK$ is the Geometric mean of $AB$ and $AQ$ and $OA$ is the arithmetic mean of $AP$ and $AQ$.

  1. Statement - 1 is true, statement -2 is true and statement -2 is correct explanation for statement -1.

  2. Statement - 1 is true, statement -2 is true and statement -2 is NOT the correct explanation for statement -1.

  3. Statement - 1 is true, statement -2 is false

  4. Statement - 1 is false , statement -2 is true
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A Correct answer
AI explanation

Let the circle's radius be R and the distance OA be d. By the geometric mean theorem in the right triangle formed by the tangent, AK^2 = AB * AQ. Since OA is the arithmetic mean of AP and AQ, and applying the right triangle altitude theorem AK^2 = AB * AQ, we can show that 1/AP + 1/AQ = 2/AB. This equation proves that AB is exactly the harmonic mean of AP and AQ, showing Statement 2 correctly explains Statement 1.

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