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February 23rd, 2018, 12:19 AM   #1
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Analytic geometry problem

Given points A(-4,-4) B (-8,-2) and C(x,0) what is x
A)If AC-CB has its greatest value
B)If AC+CB has its smallest value
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February 23rd, 2018, 01:57 AM   #2
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Using calculus, A) x = -12 and B) x = -20/3.
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February 23rd, 2018, 05:18 AM   #3
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I haven't studied calculus yet; is there another way?

Last edited by skipjack; February 23rd, 2018 at 09:28 AM.
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February 23rd, 2018, 09:12 AM   #4
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You can solve this problem with analytic geometry and a calculator:
Using the distance formula (Pythagoras):
d=(√((x2-x1)^2)+((y2-y1)^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)
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Last edited by skipjack; February 23rd, 2018 at 09:31 AM.
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February 23rd, 2018, 09:37 AM   #5
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(B) It's easy to show that the line AB intersects the x-axis at (-20/3, 0), and that this is the point C.
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