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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(17) at two points A and B.  Find (AB)^2.

 Jan 3, 2022
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first circle:

 

(x-2)2 + (y+1)= 16

 

becomes:

 

(x2 - 4x + 4) + (y2 +2y + 1) = 16

x2 - 4x + y2 + 2y + 5 = 16

x2 - 4x + y2 + 2y = 11

 

second circle:

 

(x-2)2 + (y-5)2 = 17

 

becomes:

 

x2 - 4x + 4 + y2 - 10y + 25 = 17

x2 - 4x + y2 - 10y + 29 = 17

x2 - 4x + y2 - 10y + 23 = 11

 

we have x2 - 4x + y2 + 2y = 11     and     x2 - 4x + y2 - 10y + 23 = 11

both equal 11 so they equal each other

 

x2 - 4x + y2 + 2y = x2 - 4x + y2 - 10y + 23

 

x2, -4x, y2 cancel out

 

2y = -10y +23

12y = 23

y = 23/12

 

23/12 is the y - value of both of the points where the circles intersect (A and B)

 

choose one of the equations to plug in 23/12 as y. This will find the x-values of both of the points

i choose (x-2)2 + (y+1)= 16

but you can do other one too

 

(x-2)2 + ((23/12)+1)= 16

(x-2)2 + ((23/12)+(12/12))= 16

(x-2)2 + (35/12)= 16

(x-2)2 + 1225/144 = 2304/144

(x-2)2 = 1079/144

(x-2) = sqrt(1079/144)   OR   (x-2) = -sqrt(1079/144)

x = sqrt(1079/144) + 2   OR   -sqrt(1079/144) + 2

 

These are the x values of the two points where the circle intersect

I guess AB means the line segment between them

So AB2 is the length of the line segment squared which means we subtract the two x-values then square. We dont have to worry about the y-values since they are the same for both points

 

( (sqrt(1079/144) + 2) - (-sqrt(1079/144) + 2) )2

( (sqrt(1079/144) + 2 + sqrt(1079/144) - 2) )2

( sqrt(1079/144) + sqrt(1079/144) )2

( 2sqrt(1079/144) )2

4(1079/144)

1079/36

 

or in decimal

 

29.97222....

 Jan 3, 2022

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