Question

The equation of the plane passing through the points (1, 2, 3), (–1, 4, 2) and (3, 1, 1) is ______.

Options

• 5x + y + z – 10 = 0

• 5x + y + 2z – 13 = 0

• 5x + y + 12z – 43 = 0

• 5x + 6y + 2z – 23 = 0

Answer 1

The equation of the plane passing through the points (1, 2, 3), (–1, 4, 2) and (3, 1, 1) is 5x + 6y + 2z – 23 = 0.

Explanation:

Given points are

(x₁, y₁, z₁) ≡ (1, 2, 3),

(x₂, y₂, z₂) ≡ (–1, 4, 2)

and (x₃, y₃, z₃) ≡ (3, 1, 1)

Now, required equation of the plane is given by

`|(x - x_1, y - y_1, z - z_1),(x_2 - x_1, y_2 - y_1, z_2 - z_1),(x_3 - x_1, y_3 -  y_1, z_3 - z_1)|` = 0

`\implies |(x - 1, y - 2, z - 3),(-2, 2, -1),(2, -1, -2)|` = 0

`\implies` (x – 1)(–4 – 1) – (y – 2)(4 + 2) + (z – 3)(2 – 4) = 0

`\implies` –5(x – 1) – 6(y – 2) – 2(z – 3) = 0

`\implies` 5x + 6y + 2z – 23 = 0