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Question

Two circles of radii 18cm and 8cm touch externally. The length of a direct common tangent to the two circles is

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Solution

GivenABIsthecommontangentoftwocircles,whotoucheachother.ABistheircommontangent.ThecetreofonecircleisPwithradius=8cmandthecetreofanothercircleisQwithradius=18cm.SolutionWejoinPQ,AP&PQ.AperpendicularPNisdroppedfromPtoBQatN.Now(AP&BQ)ABsincetheradiusthroughthepointofcontactofatangenttoacircleisperpendiculartothetangent.PAB=90o=QBA.AlsoPNB=PNQ=90osincePNPQ.ABNPisarectangle.SoBN=AP=8cmandPN=AB.NQ=BQBN=(188)cm=10cm.AgainPQ=(8+18)cm=26cmsincethedistancebetweenthecentresoftwocircles,touchingeachother,isthesumoftheirradii.SoΔPNQisarighttrianglewithPQashypotenuse.ByPythagorastheorem,wegetPN=PQ2NQ2=262102cm=24cm.PN=AB=24cm.AnsOptionD.
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