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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 x + y = 13.

 Jan 29, 2025
 #1
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x + y =  13 →  y = 13 - x        (1)

 

Construct a circle with a radius of 5 centered at (2,7)

 

(x - 2)^2  + (y -7)^2 =  25       (2)

 

Sub (1)  into (2)

 

(x -2)^2  + ( 13 -x -7)^2  =25

 

( x -2)^2 + ( 6 - x)^2  = 25

 

x^2 -4x + 4 + x^2 -12x + 36  =25

 

2x^2 -16x + 15  = 0

 

x = [ 16  +  sqrt [ 256 - 120] ] / 4 =   [16 + 2sqrt 34] / 4  =  4 + sqrt (34) / 2

y = 13- (4 + sqrt (34)/2)  =   9 - sqrt (34) /2

 

x = 4 -sqrt (34)/2

y = 13 - (4 -sqrt (34) /2)  =  9 + sqrt (34)/2

 

 

The points are 

 

( 4 + sqrt (34)/2 , 9 - sqrt (34)/2)   and ( 4 -sqrt (34)/2, 9 + sqrt (34) / 2)

 

cool cool cool

 Jan 29, 2025

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