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A line and a circle intersect at $A$ and $B,$ as shown below.  Find the coordinates of the midpoint of $\overline{AB}$.

 

 

The circle is x^2 + y^2 = 6.  The line is x + y = 2.

 Dec 31, 2023
 #1
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Equation of line

 

y = 2 -x

 

Sub this into equation of circle

 

x^2 + ( 2 - x)^2  = 6

 

x^2 + x^2 - 4x + 4   = 6

 

2x^2 - 4x   = 2

 

x^2 -2x  =  1

 

x^2  -2x  + 1 =  1 + 1

 

(x -1)^2   = 2       

 

x - 1  = sqrt (2)         and  x  -1 = -sqrt (2)

 

x =  1 + sqrt (2)      and  x = 1  -sqrt (2)

 

y = 1 -sqrt (2)        and y =  1 +sqrt (2)

 

 

A = (1 + sqrt (2) , 1 -sqrt (2) )          B =  (1-sqrt (2) , 1 + sqrt (2) )

 

Midpoint = (1,1)

 

 

cool cool cool

 Jan 1, 2024

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