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two circles touch externally .show that the length of common tangent (not through the point of contact of the circles)is 2*vR*vr.where R and r are the radius of the circles

two circles touch externally .show that the length of common tangent (not through the point of contact of the circles)is 2*vR*vr.where R and r are the radius of the circles

Grade:10

1 Answers

SHAIK AASIF AHAMED
askIITians Faculty 74 Points
7 years ago
Hello student,
Please find the answer to your question below
Let c1,c2 be the centres of 2 circles and r1,r2 be the radii
Then c1c2=r1+r2
The length of common tangent is t1t2
Through t1 draw t1d parallel to c1c2 to meet c2t2 at d.
Then c1c2dt1 is a parallellogram
t1d=c1c2=r1+r2
dt2=c2t2-c2d=r2-r1
t12t22=t1d2-dt22=(r1+r2)2-(r2-r1)2=4r1r2
So t1t2=2\sqrt{r1r2}

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