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Externally tangent circles with centers at points $A$ and $B$ have radii of lengths $3$ and $1,$ respectively.  A line externally tangent to both circles intersects ray $AB$ at point $C$, where $B$ is on $\overline{AC}$.  What is the length $BC$?

 Dec 30, 2023
 #1
avatar+126665 
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Let BC  = x

Triangles AEC  and BDC  are similar

BC / BD  = AC / AE

x / 1 = (4 + x) / 3

 

3x = 4 + x

2x = 4

x = 2 =  BC

 

{ DC = sqrt (BC^2 - BD^2)  = sqrt ( 2^2 - 1^2)  = sqrt (3)

tan(angle DCB)  =  - ( BD / DC)  = -1/sqrt (3)

Equation of tangent line       : y = (-1/sqrt 3) ( x -6)   }

 

cool cool cool

 Dec 31, 2023

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