Question

# Two concentric circles are of radii 5cm and 3cm. The length of the chord of the larger circle which touches the smaller circle is

A
4cm
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B
12cm
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C
2cm
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D
8cm
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Solution

## The correct option is C 8cmGiven−Oisthecentreoftwoconcentriccircles C1,theinnercirclewithradius=OP=3cmand C2,theoutercirclewithradius=OA=5cm. ABisachordofC2.IttouchesC1atP. Tofindout− ThelengthofAB=? Solution− Weknowthattheradiusthroughthepointof contactofatangenttoacircleisperpendiculartothetangent. ∴OP⊥AB⟹∠OPA=90o.........(i) Againweknowthatheperpendicular,droppedfromthecenterofacircletoanyofitschord,bisectsthelatter. ∴OPbisectsABatP. SoAB=2×AP.......(ii). NowΔOAPisarighttrianglewithOAashypotenuse (fromi).OA=5cm&OP=3cm. So,applyingPythagorastheorem,wehave AP=√OA2−OP2=√52−32cm=4cm. ∴AB=2×AP(fromii)=2×4cm=8cm.Ans−OptionD.

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