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Circles — Maths STD 10 — Question

CBSE BoardEnglish MediumSTD 10MathsCircles1 Mark
Question
Write ‘True’ or ‘False’ and justify your answer.
If angle between two tangents drawn from a point P to a circle of radius a and centre O is 60°, then $\text{OP}=\text{a}\sqrt{3}.$
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Answer

False.
PT and OT are tangent and radius respectively at contact point T. $\therefore\ \angle\text{OTP}=90^\circ$ $\Rightarrow\ \triangle\text{OTP}$ is right angle $\triangle$ at T As $\triangle\text{OPT}\cong\triangle\text{OPR}$ [By SSS criterion of congruence] $\Rightarrow\ \angle\text{OPT}=\angle\text{OPR}=\frac{1}{2}\times60^\circ=30^\circ$ $\therefore$ In right angle $\triangle\text{OPT},$ $\sin30^\circ=\frac{\text{OT}}{\text{OP}}\Rightarrow\frac{1}{2}=\frac{\text{a}}{\text{OP}}\Rightarrow\text{OP}=2\text{a}$ Hence, the given statement is false.

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