JEE Main · 2022 · 24 Jun 2022 · Shift 1 · Three Dimensional Geometry

Let a line having direction ratios

1, - 4, 2

intersect the lines

\frac{x - 7}{3} = \frac{y - 1}{- 1} = \frac{z + 2}{1}

and

\frac{x}{2} = \frac{y - 7}{3} = \frac{z}{1}

at the point A and B. Then

(AB)^{2}

is equal to ______.

Answer: 84

Solution

Let the line with direction ratios

(1, - 4, 2)

intersect the first line at

A = (7 + 3 λ, 1 - λ, - 2 + λ)

and the second line at

B = (2 μ, 7 + 3 μ, μ)

.

The direction ratios of AB are

(3 λ - 2 μ + 7, - λ - 3 μ - 6, λ - μ - 2)

, which must be proportional to

(1, - 4, 2)

\frac{3 λ - 2 μ + 7}{1} = \frac{- λ - 3 μ - 6}{- 4} = \frac{λ - μ - 2}{2}

From the first two expressions:

- 12 λ + 8 μ - 28 = - λ - 3 μ - 6 ⟹ 11 λ - 11 μ = 22 ⟹ λ - μ = - 2

From the second and third expressions:

- 2 λ - 6 μ - 12 = - 4 λ + 4 μ + 8 ⟹ 2 λ - 10 μ = 20 ⟹ λ - 5 μ = 10

Solving these two equations:

λ = - 5, μ = - 3

.

Thus

A = (- 8, 6, - 7)

and

B = (- 6, - 2, - 3)

(A B)^{2} = (- 8 + 6)^{2} + (6 + 2)^{2} + (- 7 + 3)^{2} = 4 + 64 + 16 = 84

JEE Main · 2022 · 24 Jun 2022 · Shift 1 · Integer answer