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The line $y = (x - 2)/2$ intersects the circle $x^2 + y^2 = 8$ at $A$ and $B$. Find the midpoint of $\overline{AB}$. Express your answer in the form "$(x,y)$."

michaelcai  Nov 6, 2017
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y =  (x - 2) / 2       (1)

 

x^2 + y^2   =  8    (2)

 

Sub (1)  into (2)

 

x^2  +   [  (x - 2) / 2 ]^2 =  8

 

x^2  + (1/4) (x^2 - 4x + 4 )  =  8

 

4x^2  + x^2 - 4x + 4 =  32

 

5x^2  - 4x - 28  =  0

 

(5x - 14) (x + 2)  =  0 

 

Setting each factor to 0 and solving for x we have that

 

x = 14/5      and  x   =  -2

 

The corresponding y coordinates are

 

y = (1/2) (14/5 - 2)  =  (1/2)(4/5)  = 4/10  = 2/5

 

y =  (1/2) (-2 - 2)  =  -2

 

So....the intersection  points are  ( 14/5, 2/5)  and  ( -2, -2) 

 

And the midpoint of this is  (  [ 14/5 -2] / 2 , [2/5 - 2] / 2 )  =

 

( [4/5]/2 , [-8/5]/2)  =   ( 4/10, -8/10)  =  (2/5, -4/5)

 

 

cool cool cool

CPhill  Nov 6, 2017

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