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} $