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805+ (Hard)|   Geometry|      
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Bunuel
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A circle centered at A with a radius of 1 and a circle centered at B w 18 Mar 2019, 02:53
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95% (hard)

Question Stats:

29% (03:16) correct 71% (02:44) wrong based on 38 sessions

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A circle centered at A with a radius of 1 and a circle centered at B with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is

(A) 1/3
(B) 2/5
(C) 5/12
(D) 4/9
(E) 1/2

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m1033512
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A circle centered at A with a radius of 1 and a circle centered at B w 01 Apr 2019, 04:10
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Bunuel

A circle centered at A with a radius of 1 and a circle centered at B with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is

(A) 1/3
(B) 2/5
(C) 5/12
(D) 4/9
(E) 1/2

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It can be solved by calculating the length of common tangent for all three circles


In the attached picture

Let the third circle be denoted by C and let the radius be x

common tangent LN = LM+MN

lenght of LN = √((distance between centres of circle A and B )^2 -( difference of radius of circle A and B))

= √(4+1)^2+(4-1)^2
= √25-9
=√16
= 4

Similarly we calculate LM and MN by using the above mentioned formula

now we have ,

LN = LM+MN

after putting the values of LM and MN and LN we get the below mentioned equation



4 = √(4x) +√(16x)
4 = 2√x+4√x
4 = 6√x
means x = 4/9


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A circle centered at A with a radius of 1 and a circle centered at B w 06 Apr 2019, 02:34
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m1033512

It can be solved by calculating the length of common tangent for all three circles





lenght of LN = √((distance between centres of circle A and B )^2 -( difference of radius of circle A and B))


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Hi , can you explain this please , is there a formula for this ?
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A circle centered at A with a radius of 1 and a circle centered at B w 06 Apr 2019, 04:02
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yes ,

this is formula to calculate the lenght of common tangent of circles .

You can find that in any quant book in Geometry section .

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Bunuel
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