Question

The equation of the plane through the line x + y + z + 3 = 0 = 2x – y + 3z + 1 and parallel to the line `x/1 = y/2 = z/3`, is ______.

Options

• x – 5y + 3z = 7

• x – 5y + 3z = –7

• x + 5y + 3z = 7

• x + 5y + 3z = –7

Answer 1

The equation of the plane through the line x + y + z + 3 = 0 = 2x – y + 3z + 1 and parallel to the line `x/1 = y/2 = z/3`, is x – 5y + 3z = 7.

Explanation:

Any plane through the given line

2x – y + 3z + 1 + λ(x + y + z + 3) = 0  ...(From S + λS' = 0)

If this plane is parallel to the line `x/1 = y/2 = z/3`, then the normal to the plane is also perpendicular to the above line.

∴ (2 + λ)1 + (λ – 1)2 + (3 + λ)3 = 0  ...(∵ l₁l₂ + m₁m₂ + n₁n₂ = 0)

`\implies` λ = `-3/2`

and the required plane is x – 5y + 3z – 7 = 0.