In the given figure, $PT$ is a tangent to the circle centered at $O$. $OC$ is perpendicular to chord $AB$. Prove that $PA \cdot PB = PC^2 - AC^2$.

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Question

In the given figure, $PT$ is a tangent to the circle centered at $O$. $OC$ is perpendicular to chord $AB$. Prove that $PA \cdot PB = PC^2 - AC^2$.

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Accepted Answer

Step 1: $PA \cdot PB = (PC - AC)(PC + BC)$
Step 2: $= (PC - AC)(PC + AC)$ [$AC = BC$ since perpendicular from centre bisects chord]
Step 3: $= PC^2 - AC^2$

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