PQ and RS are chords of a circle which intersect at O outside the circle. If PQ = 5 cm, OQ = 9 cm and RS = 15 cm, then the length of OS is

27 cm

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9 cm

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6 cm

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15 cm

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Solution

The correct option is C 6 cm
PQ and RS are chords whereas PQO and RSO are secant to the circle.

Using Tangent Secant Theorem
$∴ O Q \times O P = O S \times O R \Rightarrow O Q \times (O Q + P Q) = O S \times (O S + R S) \Rightarrow 9 \times (9 + 5) = O S \times (O S + 15) \Rightarrow O S^{2} + 15 (O S) - 126 = 0 \Rightarrow O S^{2} + 21 (O S) - 6 (O S) - 126 = 0 \Rightarrow O S (O S + 21) - 6 (O S + 21) = 0 \Rightarrow (O S - 6) (O S + 21) = 0 \Rightarrow O S = 6 o r O S = - 21$
(Rejecting negative value)

Thus, the length of OS is 6 cm.

Hence, the correct answer is option c.

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