A) \[\frac{2}{3}\]
B) \[\frac{1}{5}\]
C) \[\frac{3}{5}\]
D) \[\frac{2}{5}\]
Correct Answer: C
Solution :
A line makes angles\[\alpha ,\beta \]respectively an\[\gamma \]with X-axis, Y-axis and Z-axis , then \[{{\cos }^{2}}\alpha +{{\cos }^{2}}\beta +{{\cos }^{2}}\gamma =1.\] A line makes angle\[\theta \]with X-axis and Z-axis and \[\beta \]with Y-axis. \[\therefore \]\[l=\cos \theta ,\text{ }m=\cos \beta ,\text{ }n=\cos \theta \] We know that, \[{{l}^{2}}+{{m}^{2}}+{{n}^{2}}=1\] \[\Rightarrow \] \[{{\cos }^{2}}\theta +{{\cos }^{2}}\beta +{{\cos }^{2}}\theta =1\] \[\Rightarrow \] \[2{{\cos }^{2}}\theta =1-{{\cos }^{2}}\beta \] \[\Rightarrow \] \[2{{\cos }^{2}}\theta ={{\sin }^{2}}\beta \] ...(i) But \[{{\sin }^{2}}\beta =3\text{ }{{\sin }^{2}}\theta \] ...(ii) From Eqs. (i) and (ii), we get \[3{{\sin }^{2}}\theta =2{{\cos }^{2}}\theta \] \[\Rightarrow \] \[3(1-{{\cos }^{2}}\theta )=2{{\cos }^{2}}\theta \] \[\Rightarrow \] \[3=5{{\cos }^{2}}\theta \] \[\Rightarrow \] \[{{\cos }^{2}}\theta =\frac{3}{5}\]
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