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

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New post 18 Mar 2019, 02:53
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A
B
C
D
E

Difficulty:

  85% (hard)

Question Stats:

29% (02:30) correct 71% (02:46) wrong based on 17 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

Attachment:
e0340e3581aab9d635bcf2dca6b71152f8fa5290.png
e0340e3581aab9d635bcf2dca6b71152f8fa5290.png [ 20.01 KiB | Viewed 612 times ]

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

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New post 01 Apr 2019, 04:10
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Bunuel wrote:
Image
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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Re: A circle centered at A with a radius of 1 and a circle centered at B w  [#permalink]

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New post 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 ?
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Re: A circle centered at A with a radius of 1 and a circle centered at B w  [#permalink]

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New post 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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