Let P be the plane passing through the points (5, 3, 0), (13, 3, -2) and (1, 6, 2). For \(\alpha \in \mathrm{N}\), if the distances of the points \(\mathrm{A}(3,4, \alpha)\) and \(\mathrm{B}(2, \alpha, a)\) from the plane P are 2 and 3 respectively, then the positive value of \(a\) is

JEE Main · 2023 · 11 Apr 2023 · Shift 2 · Three Dimensional Geometry

Let P be the plane passing through the points (5, 3, 0), (13, 3, -2) and (1, 6, 2). For

α \in N

, if the distances of the points

A (3, 4, α)

and

B (2, α, a)

from the plane P are 2 and 3 respectively, then the positive value of

a

Answer: (B)

Two direction vectors lying in the plane are

v_{1} = (8, 0, - 2)

and

v_{2} = (- 4, 3, 2)

. Their cross product gives a normal

n = v_{1} \times v_{2} = (6, - 8, 24)

, which reduces to

(3, - 4, 12)

. Using the point

(5, 3, 0)

, the plane is

3 x - 4 y + 12 z = 3

, and

3^{2} + 4^{2} + 1 2^{2} = 13

. The distance of

A (3, 4, α)

satisfies

\frac{∣9 - 16 + 12 α - 3∣}{13} = 2

, so

∣12 α - 10∣ = 26

and

12 α = 36

, giving the natural number

α = 3

. The distance of

B (2, 3, a)

satisfies

\frac{∣6 - 12 + 12 a - 3∣}{13} = 3

, so

∣12 a - 9∣ = 39

and

12 a = 48

. Therefore the positive value of

a

4

JEE Main · 2023 · 11 Apr 2023 · Shift 2 · Single correct option