+0  
 
0
641
1
avatar+598 

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
 #1
avatar+92794 
+2

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

13 Online Users

avatar

New Privacy Policy

We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive information about your use of our website.
For more information: our cookie policy and privacy policy.