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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.

 Jun 19, 2024
 #1
avatar+129733 
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(x - 4)^2 + (y -5)^2  =  68

 

This is a circle  with a center at (4,5)

 

A perpepdicular line to the given one passing through the center amd intersecting the given line  will give us the midpoint of chord AB 

 

Equation of perpendicular line

y =(-1/3)(x - 4) + 5        set this equal to  the given line

 

3x -2 = (-1/3)(x - 4) + 5       mulriply through by 3

 

9x -6  = - (x - 4) + 15

 

10x = 25

 

x = 25/10  = 2.5

 

y = 3(2.5) -2  = 5.5

 

Midpoint =   (2.5 , 5.5)

 

cool cool cool

 Jun 19, 2024

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