The plane passing through the line $\mathrm{L}: \ell \mathrm{x}-\mathrm{y}+3(1-\ell) \mathrm{z}=1, x+2 \mathrm{y}-\mathrm{z}=2$ and perpendicular to the plane $3 x+2 y+z=6$ is $3 x-8 y+7 z=4$. If $\theta$ is the acute angle between the line L and the y-axis, then $415 \cos ^{2} \theta$ is equal to ______.

JEE Main · 2022 · 26 Jul 2022 · Shift 2 · Three Dimensional Geometry

The plane passing through the line

L : ℓ x - y + 3 (1 - ℓ) z = 1, x + 2 y - z = 2

and perpendicular to the plane

3 x + 2 y + z = 6

3 x - 8 y + 7 z = 4

. If

θ

is the acute angle between the line L and the y-axis, then

415 cos^{2} θ

is equal to ______.

Answer: 125

The direction ratios of line L are given by the cross product of the normals to the two planes:

(ℓ, - 1, 3 (1 - ℓ))

and

(1, 2, - 1)

. This gives direction ratios

(6 ℓ - 5, 3 - 2 ℓ, 2 ℓ + 1)

.

Since the plane

3 x - 8 y + 7 z = 4

contains the line, its normal

(3, - 8, 7)

is perpendicular to the line's direction vector:

3 (6 ℓ - 5) - 8 (3 - 2 ℓ) + 7 (2 ℓ + 1) = 0

18 ℓ - 15 - 24 + 16 ℓ + 14 ℓ + 7 = 0

48 ℓ = 32 \Rightarrow ℓ = \frac{2}{3}

Substituting: direction ratios are

(- 1, \frac{5}{3}, \frac{7}{3})

.

The angle with the y-axis:

cos θ = \frac{5/3}{1 + 25/9 + 49/9} = \frac{5/3}{83/9} = \frac{5}{83}

.

Therefore:

415 cos^{2} θ = 415 \times \frac{25}{83} = 125

JEE Main · 2022 · 26 Jul 2022 · Shift 2 · Integer answer