Prove that the line of section of the planes 5x + 2y – 4z + 2 = 0 and 2x + 8y + 2z – 1 = 0 parallel to the plane 4x – 2y – 5z – 2 = 0.

Let a₁,b₁ and c₁ be the direction ratios of the line 5x + 2y – 4z + 2 = 0 and 2x + 8y + 2z – 1 = 0.

As we know that if two planes are perpendicular with direction ratios as a₁, b₁ and c₁ and a₂ , b₂ and c₂ then

a₁a₂ + b₁b₂ + c₁c₂ = 0

Since, line lies in both the planes, so it is perpendicular to both planes

5a₁ + 2b₁ – 4c₁ = 0 ……(1)

2a₁ + 8b₁ + 2c₁ = 0 ……(2)

Solving equation (1) and (2) by cross multiplication we have,

⇒

⇒

⇒

∴ a = 2k, b = – k and c = 2k

We know that line is parallel to plane a₂x + b₂y + c₂z + d₂ = 0 if a₁a₂ + b₁b₂ + c₁c₂ = 0 ……(3)

Here, line with direction ratios a₁,b₁ and c₁ is parallel to plane,

4x – 2y – 5z = 5

So,

2×4 + (– 1)× – 2 + 2× – 5 = 0

⇒ 8 + 2 – 10 = 0

Therefore, the line of section is parallel to the plane.