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# A circle centered at A with a radius of 1 and a circle centered at B w

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Math Expert
Joined: 02 Sep 2009
Posts: 59588
A circle centered at A with a radius of 1 and a circle centered at B w  [#permalink]

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18 Mar 2019, 02:53
00:00

Difficulty:

85% (hard)

Question Stats:

33% (02:50) correct 67% (02:41) wrong based on 21 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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e0340e3581aab9d635bcf2dca6b71152f8fa5290.png [ 20.01 KiB | Viewed 807 times ]

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Senior Manager
Joined: 25 Feb 2019
Posts: 335
A circle centered at A with a radius of 1 and a circle centered at B w  [#permalink]

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01 Apr 2019, 04:10
2
Bunuel wrote:

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

Attachment:
The attachment e0340e3581aab9d635bcf2dca6b71152f8fa5290.png is no longer available

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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Manager
Joined: 09 Jun 2017
Posts: 104
GMAT 1: 640 Q44 V35
Re: A circle centered at A with a radius of 1 and a circle centered at B w  [#permalink]

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06 Apr 2019, 02:34
m1033512 wrote:

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 ?
Senior Manager
Joined: 25 Feb 2019
Posts: 335
Re: A circle centered at A with a radius of 1 and a circle centered at B w  [#permalink]

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06 Apr 2019, 04:02
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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Re: A circle centered at A with a radius of 1 and a circle centered at B w   [#permalink] 06 Apr 2019, 04:02
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