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Two circles intersect at two points, $P$ and $Q$. The equations of the two circles are $x^2 + (y - 1)^2 = 1$ and $(x - 1)^2 + y^2 = 1$. Find the length PQ.

 Jun 27, 2024
 #1
avatar+129725 
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Since

 

    (1)   =  (1)       set the equations  = 

 

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

 

x^2 + y^2 - 2y + 1 =  x^2 -2x + 1 + y^2

 

-2y = -2x

 

x = y

 

So

 

x^2 + ( x - 1)^2  =  1

 

x^2 + x^2 -2x + 1   = 1

 

2x^2 - 2x  =  0

 

x^2 - x = 0

 

x ( x -1) = 0

 

x = 0       x   = 1

 

y = 0      y = 1

 

The intersection points are   ( 0, 0)  and (1, 1)

 

The distance  between them =   sqrt [1^2 + 1^2]  = sqrt 2

 

 

cool cool cool

 Jun 27, 2024

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