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The circle centered at (2, -1) and with radius 4 intersects the circle centered at (2, 5) and with radius sqrt(10) at two points A and B. Find (AB)^2.

 Mar 12, 2021
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We  have  this

 

(x - 2)^2   + ( y + 1)^2  =  16

(x - 2)^2  +  ( y - 5)^2  =   10   subtract  the  second equation form the first

 

(y + 1)^2  - ( y - 5)^2   =  6        simplify

 

y^2  + 2y  + 1   -  y^2 + 10y  - 25    = 6

 

12y  - 24  =  6

 

12y  =  30  

 

y  = 30 /12   =  2.5 = 5/2

 

Using  either equation   we  can find x  as follows

 

(x - 2)^2 + ( 2.5 + 1)^2   = 16

(x - 2)^2  +  12.25 =   16

(x - 2)^2  =  16 -12.25

(x  -2)^2  =  15/4         take both roots

 

x - 2  =  sqrt (15)/2                              x   -  2  = -sqrt (15) /2

x =  sqrt (15)  / 2  +  2                        x  =  2 - sqrt (15) /2

x=  [ 4  +  sqrt (15)  ]   /2                    x=  [4 -sqrt (15)]  / 2

 

 

A = (  [4 + sqrt (15) ] / 2 , 5/2 )

B = ( [4 - sqrt (15) ]   / 2 , 5/2 )

 

My question is.....  What is AB  supposed to represent ????

 

 

cool cool cool

 Mar 12, 2021
edited by CPhill  Mar 12, 2021

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