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Question

Length of internal (transverse) common tangent to two circles is Lint=d2(r1r2)2

Where d = distance between the centers of the two circles r1& r2 are radii of two circles


A

True

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B

False

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Solution

The correct option is B

False


Let center of two circles being c1& c2 and their radii r1& r2

Then,

Given,

c1 c2 = d

Let transverse tangent touches the circle at A and B. It intersects the line segment c1 c2 at P.

Let length of transverse tangent AB = Ltrans

Right angled triangle c1 AP

C1P2 = r22+ AP2- - - - - - (1)

Right angles triangle C2BP

C2P2 = r22+ PB2- - - - - - (2)

In AC1P & BC2P

C1AP = C2BP = 90

APc1 = BPc2 = oppositeangle

Since, two angles of a triangle are equal, third angle should also be equal. {sum of the angle of triangle = 180

AC1P BC2P {AAA similarity}

Ac1Bc2=ApBp=c1pc2p

r2r1=LtransBPBP=dC2PC2P______r2r1=Ltransr1r1+r2r2r1=dC2PC2P C2P=dr1r1+r2

Substituting the values of BP & c2P in equation (2)

d2r12(r1+r2)2=r12+L2transr12(r1+r2)2

d2=(r1+r2)2+L2trans

L2trans=d2(r1+r2)2

Ltrans=d2(r1+r2)2

So,given statment

Length of internal (transverse) common tangent to circles is

Lint=d2(r1r2)2 is NOT correct.


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