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I have next problem:
Find the point $C$ on the $Oxy$ plane, the sum of the distances from it to the points $A(-7,3,5)$ and $B(2,-2,3)$ would be the smallest.

There is my progress:
Let us assume that the $C_1$ point isn't fixed in $Oxy$ plane, then it would lay on the line $AB$ in between points $A$ and $B$. Then I "assumed" that point $C_1$ is in the center of $AB$ and found its coordinates $(-2.5,0.5,4)$ with segment dividing formula. The ruler measures say that the point $C_1$ projected on the $Oxy$ plane $(-2.5,0.5,0)$ has the properties from the problem description. BUT I don't get it. Why point $C_1$ projected on the $Oxy$ plane is the $C$ point? I can prove it with the multivariable calculus but not the analytic geometry way. Can you help me figure this one geometry solution out?

Andrei
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Vladyslav Rehan
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1 Answers1

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Let's choose a point $D=(x_0,y_0,0)$ in the $0xy$ plane, and reflect point $B$ with respect to same plane, to $B'=(2,-2,-3)$. Then you can easily show that $DB=DB'$. Then $$AD+DB=AD+DB'$$ To minimize this sum, $D$ is at the intersection of the $AB'$ line and the $0xy$ plane. You can write this as $$\frac{x_A-x}{x_A-x_B}=\frac{y_A-y}{y_A-y_B}=\frac{z_A-0}{z_A+z_B}$$ In the last term I used $z_{B'}=-z_B$. You can now get $x$ and $y$.

Andrei
  • 37,370