JEE Main · 2023 · 01 Feb 2023 · Shift 2 · Three Dimensional Geometry

Let

α x + β y + γ z = 1

be the equation of a plane passing through the point

(3, - 2, 5)

and perpendicular to the line joining the points

(1, 2, 3)

and

(- 2, 3, 5)

. Then the value of

α β γ

is equal to ______.

Answer: 6

Solution

Because the plane is perpendicular to the line joining

(1, 2, 3)

and

(- 2, 3, 5)

, that line lies along the plane's normal. Its direction is

(- 2 - 1, 3 - 2, 5 - 3) = (- 3, 1, 2)

, equivalently

(3, - 1, - 2)

. Writing the plane as

α x + β y + γ z = 1

, the normal

(α, β, γ)

is proportional to this direction; testing the point

(3, - 2, 5)

with

(α, β, γ) = (3, - 1, - 2)

gives

3 (3) + (- 1) (- 2) + (- 2) (5) = 9 + 2 - 10 = 1

, which satisfies the equation, so

α = 3, β = - 1, γ = - 2

. Hence

α β γ = (3) (- 1) (- 2) = 6

JEE Main · 2023 · 01 Feb 2023 · Shift 2 · Integer answer