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Find all points $(x,y)$ that are $5$ units away from the point $(2,7)$ and that lie on the line $y = 5x - 28.$

 Mar 15, 2024
 #1
avatar+128794 
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All points 5 units away from (2,7) can be  represented by a circle with the following equation

 

(x - 2)^2 + (y -7)^2    =  25

 

find the intersection  of this  circle with the line thusly

 

(x -2)^2  + (5x -28 - 7)^2  =  25

 

(x - 2)^2 + (5x - 35)^2  = 25

 

x^2 - 4x + 4  + 25x^2 - 350x + 1225  = 25

 

26x^2 - 354x  + 1204  =  0

 

x =  86/13        and   x  = 7

 

y = 5(86/13) - 28  = 66/13                      y =  5(7) - 28 =  7

 

The points are   ( 86/13 , 66/13)  and ( 7 , 7)

 

 

cool cool cool

 Mar 15, 2024

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