JEE Main · 2025 · 03 Apr 2025 · Shift 2 · Three Dimensional Geometry

Each of the angles

β

and

γ

that a given line makes with the positive y - and z -axes, respectively, is half of the angle that this line makes with the positive

x

-axes. Then the sum of all possible values of the angle

β

Answer: (A)

Solution

Let the line make angle

α

with the positive

x

-axis, so

β = γ = \frac{α}{2}

. The direction cosines satisfy

cos^{2} α + cos^{2} β + cos^{2} γ = 1

, giving

cos^{2} α + 2 cos^{2} \frac{α}{2} = 1

. Since

2 cos^{2} \frac{α}{2} = 1 + cos α

, this becomes

cos^{2} α + cos α = 0

, so

cos α (cos α + 1) = 0

. Hence

α = \frac{π}{2}

α = π

, giving

β = \frac{π}{4}

β = \frac{π}{2}

. The sum of all possible values of

β

\frac{3 π}{4}

JEE Main · 2025 · 03 Apr 2025 · Shift 2 · Single correct option