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

JEE Main · 2004 · 24 Apr 2004 · Offline · Three Dimensional Geometry

A line makes the same angle

θ

, with each of the

x

and

z

axis. If the angle

β

, which it makes with

y

-axis, is such that

sin^{2} β = 3 sin^{2} θ

, then

cos^{2} θ

equals

(A) $\frac{2}{3}$
(B) $\frac{1}{5}$
(C) $\frac{3}{5}$ ✓ Correct
(D) $\frac{2}{5}$

Answer: (C)

Solution

With direction cosines

l = cos θ

and

n = cos θ

for the equal angles with the

x

- and

z

-axes, and

m = cos β

, the identity

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

gives

2 cos^{2} θ + cos^{2} β = 1

. The condition

sin^{2} β = 3 sin^{2} θ

rewrites as

1 - cos^{2} β = 3 (1 - cos^{2} θ)

, so

cos^{2} β = 3 cos^{2} θ - 2

. Substituting,

2 cos^{2} θ + 3 cos^{2} θ - 2 = 1

, hence

5 cos^{2} θ = 3

and

cos^{2} θ = \frac{3}{5}

JEE Main · 2004 · 24 Apr 2004 · Offline · Single correct option

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