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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.

 Apr 1, 2024
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x + y =  13

y = 13 - x

 

We have a circle with a radius of  5 and a  center of (2,7).....the equation is

 

(x - 2)^2  + (y - 7)^2  =   25

 

(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 + 25  =  0 

 

x =  [ 16 ±  sqrt [ 256 - 200] ] /  4  =   [ 16± sqrt [ 56 ] ] /  4 =   

 

x =  [ 8 + sqrt 14 ] / 2      and  x =  [ 8  - sqrt 14 ] ] / 2

 

x = 4 + sqrt [14 ] / 2        and x = 4  -sqrt [14 ] / 2

 

y  = 13 - x

 

y =  9 - sqrt [14] / 2           and  y =   9 + sqrt [14] / 2

 

 

The points are   ( 4 + sqrt [ 14] / 2 , 9 - sqrt [14] / 2)   and (4 -sqrt [14] / 2 , 9 + sqrt [14]  / 2)

 

cool cool cool

 Apr 1, 2024

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