Question

The projection of a line segment $OP$ through origin $O$, on the coordinate axes are $8,5,6$. Then, the length of the line segment $OP$ is equal to

• A. $5$
• B. $5\sqrt{5}$
• C. $10\sqrt{5}$
• D. None of these

Answer 1

Answer: (B)

Let $l,$ $m$ and $n$ be the direction cosine's of the given line segment PQ.
$\therefore$ $l=\cos \alpha ,m=\cos \beta ,n=\cos \gamma$
where $\alpha ,\beta ,\gamma$
are the angles which the line segment PQ makes with the axes. Suppose length of line segment
$PQ=r$
This, projection of line segment PQ on x-axis
$=PQ\cos \alpha =rl$
Also, the projection of line segment PQ on x-axis
$=8$
$\therefore$ $lr=8$
Similarly $mr=5,nr=6$
Now, on squaring adding these equations, we get
${(lr)}^2+{(mr)}^2+{(nr)}^2=8^2+5^2+6^2$
$r^2(l^2+m^2+n^2)=64+25+36$
$(\because l^2+m^2+n^2=1)$
$\Rightarrow$ $r^2=125$
$\Rightarrow$ $r=5\sqrt{3}$