February 23rd, 2018, 12:19 AM  #1 
Newbie Joined: Feb 2018 From: Afghanistan Posts: 17 Thanks: 0  Analytic geometry problem
Given points A(4,4) B (8,2) and C(x,0) what is x A)If ACCB has its greatest value B)If AC+CB has its smallest value 
February 23rd, 2018, 01:57 AM  #2 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,859 Thanks: 1080 Math Focus: Elementary mathematics and beyond 
Using calculus, A) x = 12 and B) x = 20/3.

February 23rd, 2018, 05:18 AM  #3 
Newbie Joined: Feb 2018 From: Afghanistan Posts: 17 Thanks: 0 
I haven't studied calculus yet; is there another way?
Last edited by skipjack; February 23rd, 2018 at 09:28 AM. 
February 23rd, 2018, 09:12 AM  #4 
Newbie Joined: Dec 2017 From: Spain Posts: 18 Thanks: 1 
You can solve this problem with analytic geometry and a calculator: Using the distance formula (Pythagoras): d=(√((x2x1)^2)+((y2y1)^2) We have that the distances AC and CB are: AC=[√((x(4))^2)+(4)^2]=[√(x^2)+8x+32] CB[√((x(8))^2)+(2)^2]=[√(x^2)+16x+68] Hence: AC±CB=[√(x^2)+8x+32]±[√(x^2)+16x+68] which are two different functions. With the calculator, you can get the highest point of the one function and the lowest point of the other function: The results will be: Highest point: x = 12 Lowest point: x = (20/3) Last edited by skipjack; February 23rd, 2018 at 09:31 AM. 
February 23rd, 2018, 09:37 AM  #5 
Global Moderator Joined: Dec 2006 Posts: 19,527 Thanks: 1750 
(B) It's easy to show that the line AB intersects the xaxis at (20/3, 0), and that this is the point C.


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