A line makes the same angle \Theta , with each of the x\; and\; z\; axis. If the angle \beta , which it makes with y-axis, is such that \sin ^{2}\beta =3\sin ^{2}\Theta ,then\; \cos ^{2}\Theta equals:

  • Option 1)

    3/5

  • Option 2)

    1/5

  • Option 3)

    2/3

  • Option 4)

    2/5

 

Answers (1)

As we learnt in

Direction Cosines -

If \alpha, \beta, \gamma are the angles which a vector makes with positive X-axis,Y-axis and Z-axis respectively then\cos \alpha, \cos \beta, \cos \gamma are known as diresction cosines, generally denoted by (l,m,n).

l=\cos \alpha, m=\cos \beta, n=\cos \gamma

l^{2}+m^{2}+n^{2}= 1

\cos^{2} \alpha+ \cos^{2} \beta+\cos^{2} \gamma= 1

- wherein

 

  Given \alpha =\gamma= \theta

Also, \sin ^{2}\beta = 3\sin ^{2}\theta

1- \cos ^{2}\beta = 3- 3\cos ^{2}\theta

\cos ^{2}\beta = 3\cos ^{2}\theta-2

\cos ^{2}\alpha +\cos ^{2}\beta +\cos ^{2}\gamma = 1

5\cos ^{2}\theta -2 = 1

\cos ^{2}\theta= \frac{3}{5}

 


Option 1)

3/5

Correct option

Option 2)

1/5

Incorrect option

Option 3)

2/3

Incorrect option

Option 4)

2/5

Incorrect option

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