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A line and a circle intersect at A and B.  Find the coordinates of the midpoint of \overline{AB}.

 

The ilne is y = 3x - 2, and the circle is (x - 4)^4 + (y - 5)^2 = 68.

 Dec 23, 2023
 #1
avatar+129471 
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Sub the equation of the line into the equation of the  circle

 

(x - 4)^2 + (3x - 2 -5)^2  = 68

 

(x -4)^2  + ( 3x - 7)^2  = 68

 

x^2 - 8x + 16 + 9x^2 -42x + 49   = 68

 

10x^2 - 50x - 3   = 0

 

Using the Quadratic Formula the solutions  for  x  are

 

x = (5/2) + sqrt (131/5) / 2

x = (5/2) - sqrt (131/5) / 2

 

Adding  these values and  dividing  by 2 gives us  the  x coordinate of the  midpoint  =  5/2

 

The y values  are    3 (5/2 + sqrt (131/5)  / 2)  - 2    =   15/2  + 3sqrt (131/5)/2 - 2

And  3(5/2 - sqrt (131/5) /2) -2 =  15/2 - 3sqrt (131/5)/2 - 2

 

Adding these and  dividing by 2 gives us   the y coordinate of the  midpoint = 11/2

 

Midpoint = (5/2, 11/2)

 

 

cool cool cool

 Dec 23, 2023

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