A line makes the same angle θ with each of the X and Z -axes. If the angle β , which it makes with Y-axis, is such that ^ 2 β=3 ^ 2 θ, then ^ 2 θ equals

Correct answer: C. 3 / 5 Let l, ~m and n be the direction cosines. Then, l= θ, m= β, n= θ array l we have l^ 2 +m^ 2 +n^ 2 =1 ^ 2 θ+ ^ 2 β+ ^ 2 θ=1 2 ^ 2 θ+1- ^ 2 β=1 2 ^ 2 θ- ^ 2 β=0 2 ^ 2 θ-3 ^ 2 β=0 [ ^ 2 β=3 ^ 2 0( given )] ^ 2 θ=2 / 3 ^ 2 θ= 1 1+ ^ 2 θ =(1)/(1+2 / 3)=(3)/(5) array

BITSAT 2015

Exam: BITSAT
Year: 2015
Subject: Mathematics
Chapter: Three Dimensional Geometry

BITSAT 2015 · Mathematics · Three Dimensional Geometry

A line makes the same angle θ with each of the X and Z -axes. If...

A line makes the same angle

\theta

with each of the

\mathrm{X}

and

\mathrm{Z}

-axes. 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

Options

A $2 / 5$
B $1 / 5$
C $3 / 5$
D $2 / 3$

Correct answer

C. $3 / 5$

Step-by-step solution

Let

l, \mathrm{~m}

and

\mathrm{n}

be the direction cosines. Then,

l=\cos \theta, \mathrm{m}=\cos \beta, \mathrm{n}=\cos \theta

\begin{array}{l}\text{ we have }l^{2}+m^{2}+n^{2}=1 \\ \Rightarrow \cos ^{2} \theta+\cos ^{2} \beta+\cos ^{2} \theta=1 \\ \Rightarrow 2 \cos ^{2} \theta+1-\sin ^{2} \beta=1 \\ \Rightarrow 2 \cos ^{2} \theta-\sin ^{2} \beta=0 \\ \Rightarrow 2 \cos ^{2} \theta-3 \sin ^{2} \beta=0 \\ \quad\left[\because \sin ^{2} \beta=3 \sin ^{2} 0(\text{ given })\right]\\ \Rightarrow \tan ^{2} \theta=2 / 3 \\ \therefore \cos ^{2} \theta=\frac{1}{1+\tan ^{2} \theta}=\frac{1}{1+2 / 3}=\frac{3}{5} \end{array}

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