Question

A $=(1,1,4)$ and B $=(5,-3,4)$ are two points. If the points $P$, $Q$ are on the line AB such that AP $=$ PQ $=$ QB then PQ $=$

• A. $2\sqrt{2}$
• B. $4$
• C. $\sqrt{\frac{32}{9}}$
• D. $\sqrt{2}$

Answer 1

The correct option is C $\sqrt{\frac{32}{9}}$

$AB=\sqrt{(1-5{)}^2+(1+3{)}^2+(4-4{)}^2}$

$AB=\sqrt{(-4{)}^2+4^2}$

$AB=\sqrt{32}$

Given that $AP=PQ=QB$

But, $AB=AP+PQ+QB$

Hence, $AB=3\times PQ$

So, $PQ=\frac{\sqrt{32}}{3}=\sqrt{\frac{32}{9}}$