A) \[\frac{2}{3}\]
B) \[\frac{1}{5}\]
C) \[\frac{3}{5}\]
D) \[\frac{2}{5}\]
Correct Answer: C
Solution :
A line makes angle\[\theta \]with\[x-\]axis and z-axis and \[\beta \]withy-axis. \[\therefore \] \[l\cos \theta ,m=\cos \beta ,n=\cos \theta \] 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 \[{{\sin }^{2}}\beta =3{{\sin }^{2}}\beta \] ...(ii) \[\therefore \]From Eqs. (i) and (ii) \[3{{\sin }^{2}}\theta =2{{\cos }^{2}}\theta \] \[\Rightarrow \] \[3(1-{{\cos }^{2}}\theta )=2{{\cos }^{2}}\theta \] \[\Rightarrow \] \[3-3{{\cos }^{2}}\theta =2{{\cos }^{2}}\theta \] \[\Rightarrow \] \[3=5{{\cos }^{2}}\theta \] \[\Rightarrow \] \[{{\cos }^{2}}\theta =\frac{3}{5}\]
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