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The circle centered at (2,-1) and with radius 4 intersects the circle centered at (2,6) and with radius sqrt(18) at two points A and B. Find (AB)^2.

 Jun 28, 2022
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The equations for  the circles are

 

(x - 2)^2 + ( y + 1)^2  = 16        (1)

(x -2)^2 + ( y - 6)^2 =    18       (2)           subtract    (1) from (2)

 

(y - 6)^2 - (y + 1)^2  =  2

 

y^2 - 12y + 36 - y^2 - 2y - 1  =  2

 

-14y + 35 =  2

 

-14y = -33

 

y = -33 /-14  =  33/14      this is the y coordinate for both points

 

To find the x coordinates

 

(x  -2)^2  + (33/14 + 1)^2   = 16

 

(x - 2)^2  +  ( 47/14)^2  =16

 

(x - 2)^2  =   16 - (47/14)^2

 

( x-2)^2  =    927 / 196        take both roots

 

x - 2 =   sqrt (927/196)       or        x  - 2  =  -sqrt (927/16)

 

x = sqrt (927/16) + 2       or           x =  -sqrt (927 /196)  + 2

 

AB  =     [ sqrt (927/16) + 2 ]  -  [ sqrt (927 / 196)  + 2 ]    =   2 sqrt (927 /196)

 

(AB)^2  =  4 ( 927 / 196)  =  927 / 49

 

 

 

cool cool cool

 Jun 28, 2022

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