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Two circles with radii 'a' and 'b' touch each other at a point. A common direct common...

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nasirjam | Student, Grade 9 | (Level 2) Honors

Posted April 29, 2013 at 10:51 AM via web

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Two circles with radii 'a' and 'b' touch each other at a point. A common direct common tangent is drawn to both the circles. Another small circle is nested between the two circles and the tangent such that it just touches the circles and the tangent. The radius of the smaller is c. Prove that 1/sqrt(c)  = 1/sqrt(a) + 1/sqrt(b)

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pramodpandey | College Teacher | (Level 3) Valedictorian

Posted April 29, 2013 at 12:43 PM (Answer #1)

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Join cntres of the cicle say C(P,a), C(Q,b) and C(O,c).Let b<a.Let M and N are on the circle C(P,a) and C(Q,b) through which direct common tangent passes through. Draw perpendiculars from O to PM and PN say OL and OK.Also draw perpendicular from Q to PM say QT.

From this we have 

MN=QT=OK

OT=b-a

OP=a+c

OQ=b+c

PL=a-c

QK=b-c  Thus

`OL=sqrt((a+c)^2-(a-c)^2)`

`OL=sqrt(4ac)`

`OK=sqrt((b+c)^2-(b-c)^2)=sqrt(4bc)`

Thus  QT=OL+OK=`sqrt(4ac)+sqrt(4bc)`

In trianle PQT

QT=`sqrt((a+b)^2-(a-b)^2)=sqrt(4ab)`

Thus

`sqrt(4ac)+sqrt(4bc)=sqrt(4ab)`

divide by `sqrt(4abc)`  ,we have

`1/sqrt(c)=1/sqrt(a)+1/sqrt(b)`

 

 

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nasirjam | Student, Grade 9 | (Level 2) Honors

Posted April 30, 2013 at 10:00 AM (Answer #2)

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Do the points M and N lie on the common tangent / lie on any other point on the circle ?

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