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Need help here

 

Find all points (x,y) that are 5 units away from the point (2,7) and that lie on the line x + y = 13.

 Dec 24, 2023
 #1
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What we want  is the   intersection of the  circle with the  equation

(x -2)^2 + (y - 7)^2 =  25      (1)

with the line  y = 13 - x     (2)

 

Sub  (2)  into (1)

(x - 2)^2  +  (13 - x - 7)^2  =  25

(x - 2)^2  + (6 -x)^2   = 25

x^2 - 4x + 4 + x^2 -12x + 26 = 25

2x^2 -8x + 30  = 25

2x^2 -8x + 5  = 0

 

The  x values are

 

[ 8 + sqrt ( 8^2 - 4*2 *5) ] / 4   =   [ 8 + sqrt (24) ] / 4  =  2  +  sqrt (6) / 2 

And 2 - sqrt (6) / 2

 

The y values are   13 - [ 2 + sqrt (6)/2]  =   11 - sqrt (6)/2

And   13 - [2 - sqrt (6)/2] =    11 + sqrt (6) / 2

 

So   the points are

 

( 2 + sqrt (6)/2 , 11 - sqrt (6) /2)     and ( 2 -sqrt (6)/2 , 11 + sqrt (6) /2)

 

 

cool cool cool

 Dec 24, 2023

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