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There's A(-2,-1) and B(-4,2), the dot P is on the X axis and the absolute value of the line PA minus the line PB is the biggest. Where is dot P?
 

yomyhomies  Apr 12, 2017

Best Answer 

 #2
avatar+79741 
+1

Let P be denoted as  (x, 0)

 

So PA  =    sqrt [ ( -2 -x)^2  + (-1)^2 ]    =  sqrt [ 4 + 4x + x^2 + 1]  =  sqrt [ x^2 + 4x + 5]

 

And PB  =  sqrt [ (-4 -x)^2  + 2^2 ]  =  sqrt [ 16 + 8x + x^2 + 4 ]  = sqrt [ x^2 + 8x + 20 ]

 

And we wish to maximize D.....where D  =  

 

abs [sqrt(x^2 + 4x + 5)  -  sqrt ( x^2 + 8x + 20) ]    

 

Take the derivative of the function inside the absolute value bars and and set to 0

 

[x + 2] / sqrt (x^2 + 4x + 5)   -   [x + 4] /  sqrt (x^2 + 8x + 20)    = 0

 

[x + 2] / sqrt (x^2 + 4x + 5)   =   [x + 4] /  sqrt (x^2 + 8x + 20)    cross-multiply

 

[x + 2]sqrt(x^2 + 8x + 20)   =  [x + 4]sqrt(x^2 + 4x+ 5)        square both sides

 

[x^2 + 4x + 4] [x^2 + 8x + 20]   = [x^2 + 8x + 16] [x^2 + 4x + 5]   simplify

 

x^4 + 12 x^3 + 56 x^2 + 112 x + 80   = x^4 + 12 x^3 + 53 x^2 + 104 x + 80

 

56x^2 + 112x   =  53x^2  + 104x

 

3x^2 + 8x  =  0       factor

 

 x (3x + 8) = 0

 

There are two possible solutions......x  = 0    or x  = -8/3

 

And this graph confirms that for the original function,

abs [sqrt(x^2 + 4x + 5)  -  sqrt ( x^2 + 8x + 20) ], 

the absolute value of PA -PB  is greatest ( = √5 ) when  x =0  

 

https://www.desmos.com/calculator/vvmfqwf6wv

 

So.....(0,0)  is the point that maximizes the absolute value of PA - PB

 

{Thanks to heureka for pointing out my previous mistake...]

 

 

 

cool cool cool

CPhill  Apr 12, 2017
edited by CPhill  Apr 13, 2017
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2+0 Answers

 #2
avatar+79741 
+1
Best Answer

Let P be denoted as  (x, 0)

 

So PA  =    sqrt [ ( -2 -x)^2  + (-1)^2 ]    =  sqrt [ 4 + 4x + x^2 + 1]  =  sqrt [ x^2 + 4x + 5]

 

And PB  =  sqrt [ (-4 -x)^2  + 2^2 ]  =  sqrt [ 16 + 8x + x^2 + 4 ]  = sqrt [ x^2 + 8x + 20 ]

 

And we wish to maximize D.....where D  =  

 

abs [sqrt(x^2 + 4x + 5)  -  sqrt ( x^2 + 8x + 20) ]    

 

Take the derivative of the function inside the absolute value bars and and set to 0

 

[x + 2] / sqrt (x^2 + 4x + 5)   -   [x + 4] /  sqrt (x^2 + 8x + 20)    = 0

 

[x + 2] / sqrt (x^2 + 4x + 5)   =   [x + 4] /  sqrt (x^2 + 8x + 20)    cross-multiply

 

[x + 2]sqrt(x^2 + 8x + 20)   =  [x + 4]sqrt(x^2 + 4x+ 5)        square both sides

 

[x^2 + 4x + 4] [x^2 + 8x + 20]   = [x^2 + 8x + 16] [x^2 + 4x + 5]   simplify

 

x^4 + 12 x^3 + 56 x^2 + 112 x + 80   = x^4 + 12 x^3 + 53 x^2 + 104 x + 80

 

56x^2 + 112x   =  53x^2  + 104x

 

3x^2 + 8x  =  0       factor

 

 x (3x + 8) = 0

 

There are two possible solutions......x  = 0    or x  = -8/3

 

And this graph confirms that for the original function,

abs [sqrt(x^2 + 4x + 5)  -  sqrt ( x^2 + 8x + 20) ], 

the absolute value of PA -PB  is greatest ( = √5 ) when  x =0  

 

https://www.desmos.com/calculator/vvmfqwf6wv

 

So.....(0,0)  is the point that maximizes the absolute value of PA - PB

 

{Thanks to heureka for pointing out my previous mistake...]

 

 

 

cool cool cool

CPhill  Apr 12, 2017
edited by CPhill  Apr 13, 2017

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