A) \[3/5\]
B) \[1/5\]
C) \[2/5\]
D) \[2/3\]
Correct Answer: A
Solution :
Given line makes the same angle \[\theta \] with each of the x and z-axis and angle \[\beta \] with y-axis \[\therefore \] \[l=\cos \theta ,m=\cos \beta \] and \[n=\cos \theta \] Now, \[{{l}^{2}}+{{m}^{2}}+{{n}^{2}}=1\] \[\Rightarrow \] \[{{\cos }^{2}}\theta +{{\cos }^{2}}\beta +{{\cos }^{2}}\theta =1\] \[\Rightarrow \] \[2{{\cos }^{2}}\theta +{{\cos }^{2}}\beta =1\] \[\Rightarrow \] \[2co{{a}^{2}}\theta =1-{{\cos }^{2}}\beta ={{\sin }^{2}}\beta \] \[(\because \,\,{{\cos }^{2}}\theta +{{\sin }^{2}}\theta =1)\] \[\Rightarrow \] \[2{{\cos }^{2}}\theta ={{\sin }^{2}}\beta \] Also given, \[{{\sin }^{2}}\beta =3{{\sin }^{2}}\theta \] \[\therefore \] \[2{{\cos }^{2}}\theta =3{{\sin }^{2}}\theta =3(1-{{\cos }^{2}}\theta )\] \[\Rightarrow \] \[5{{\cos }^{2}}\theta =3\,\,\,\Rightarrow \,\,\,{{\cos }^{2}}\theta =\frac{3}{5}\]
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