Question

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

Answer 1

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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ANSWER

Given−

AB Is thecommon tangentoftwocircles,who touch

eachother.

ABistheircommontangent.

ThecetreofonecircleisPwithradius=8cm

andthecetreofanothercircleisQwithradius=18cm.

Solution−

WejoinPQ,AP&PQ.

AperpendicularPNisdroppedfromPtoBQatN.

Now(AP&BQ)⊥ABsincetheradiusthroughthepoint

ofcontactofatangenttoacircleisperpendicular

tothetangent.

∴∠PAB=90

o

=∠QBA.

Also∠PNB=∠PNQ=90

o

sincePN⊥PQ.

∴ABNPisarectangle.

SoBN=AP=8cmandPN=AB.

∴NQ=BQ−BN=(18−8)cm=10cm.

AgainPQ=(8+18)cm=26cmsincethedistancebetweenthecentres

oftwocircles,touchingeachother,isthesumoftheirradii.

SoΔPNQisarighttrianglewithPQashypotenuse.

∴ByPythagorastheorem,weget

PN=

PQ

2

−NQ

2

=

26

2

−10

2

cm=24cm.

∴PN=AB=24cm.

Ans−OptionD.

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