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Given points $A(-4,-4), B (-8,-2)$ and $C(x,0)$, what is $x$

  1. If $AC-CB$ has its greatest value
  2. If $AC+CB$ has its smallest value

I know that if $A$ and $B$ and $C$ are collinear $AC-BC$ has its greatest value so $x$ is $-12$ and $AC+BC$ also has its smallest value when they points $A,B,C$ are collinear so $x$ must be $-12$ again but answer is $-20/3$ why is this so I think this question might be solved using calculus which I haven't studied yet is there any way I can solve this without calculus?

krirkrirk
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Marva Jami
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1 Answers1

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HINT: $$AC-BC$$ is given by $$\sqrt{(x+4)^2+16}-\sqrt{(x+8)^2+4}$$ and $$AC+BC=\sqrt{(x+4)^2+16}+\sqrt{(x+8)^2+4}$$ and $$AC-BC\le 2\sqrt{5}$$ for $x=-12$ and $$AC+BC\geq 2\sqrt{13}$$ for $$x=-\frac{20}{3}$$

Dr. Sonnhard Graubner
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