A) 3
B) 4
C) 2
D) None of these
Correct Answer: D
Solution :
Let l, m and n be the direction cosines of a line. Since, the line is equally inclined with OX, OY and OZ. \[\therefore \] \[l=m-n\] \[(\because \,\,\,\cos \,\,\alpha =\cos \beta =\cos \gamma )\] Now, \[{{l}^{2}}+{{m}^{2}}+{{n}^{2}}=1\Rightarrow 3{{l}^{2}}=1\] \[\Rightarrow \] \[{{l}^{2}}=1/3\] \[\Rightarrow \] \[l=\pm \frac{1}{\sqrt{3}}\] Hence, the direction cosines of given line are \[\pm \frac{1}{\sqrt{3}},\,\,\pm \frac{1}{\sqrt{3}},\,\,\pm \,\frac{1}{\sqrt{3}},\] Since, \[+ve\] and \[-ve\] signs can be arranged at three places in \[2\times 2\times 2=8\] ways. Therefore, there are eight lines which are equally inclined with the coordinate axis.
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