Find the distance between the planes 2x + y - 2z + 6 = 0 and 4x + 2y - 4z - 6 = 0 ?

Concept:

The distance between two parallel planes ax + by + cz +d₁ = 0 and ax + by + cz +d2 = 0 is given by: \(D= \left | \frac{d_{1}-d_{2}}{\sqrt{a^2+b^2+c^2}} \right |\)

Calculation:

Given: 2x + y - 2z + 6 = 0 and 4x + 2y - 4z - 6 = 0 are two planes.

Here, we can rewrite the equation of plane 2x + y - 2z + 6 = 0 as 4x + 2y - 4z + 12 = 0 by multiplying both the sides of 2x + y - 2z + 6 = 0 with 2.

As we can see that, the plane 4x + 2y - 4z + 12 = 0 and 4x + 2y - 4z - 6 = 0 are parallel planes.

As we know that, the distance between two parallel planes ax + by + cz +d1 = 0 and ax + by + cz +d2 = 0 is given by: \(D= \left | \frac{d_{1}-d_{2}}{\sqrt{a^2+b^2+c^2}} \right |\)

Here, a = 4, b = 2, c = - 4, d₁ = 12 and d₂ = - 6.

​⇒ \(D= \left | \frac{d_{1}-d_{2}}{\sqrt{a^2+b^2+c^2}} \right |\)​

⇒ \(D= \left | \frac{12-(-6)}{\sqrt{4^2+2^2+(-4)^2}} \right |\)

⇒ ​\(D= \left | \frac{18}{\sqrt{16+4+16}} \right |\)

⇒ \(D= \left | \frac{18}{6} \right |=3\)

Hence, option 4 is correct.