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avatar+1999 

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 9, 2024
 #1
avatar+729 
0

PQ = 2*sqrt(5)

 Jun 9, 2024
 #2
avatar+129747 
+1

Since both of the radiuses = 1, set these circles =

 

x^2  + (y-1)^2  = (x -1)^2 + y^2

 

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

 

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

 

PQ =   sqrt [1 + 1]  =  sqrt 2

 

cool cool cool

 Jun 9, 2024
edited by CPhill  Jun 9, 2024

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