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)

$5\sqrt{5}$

Answer 2

Let $l,$ $m$ and $n$ be the direction cosine's of the given line segment P
$\therefore$ $l=\cos \,\alpha ,\,\,\,m=\cos \beta ,\,\,n=\cos \,\gamma$
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
${{(lr)}^{2}}+{{(mr)}^{2}}+{{(nr)}^{2}}={{8}^{2}}+{{5}^{2}}+{{6}^{2}}$
${{r}^{2}}({{l}^{2}}+{{m}^{2}}+{{n}^{2}})=64+25+36$
$(\because \,\,\,\,{{l}^{2}}+{{m}^{2}}+{{n}^{2}}=1)$
$\Rightarrow$ ${{r}^{2}}=125$
$\Rightarrow$ $r=5\sqrt{3}$