The distance between the parallel planes 4x – 2y + 4z + 9 = 0 and 8x – 4y + 8z + 21 = 0 is

Concept:

The distance between two parallel planes ax + by + cz + d₁ = 0 and ax + by + cz + d₂ = 0 is given by: \(\left| {\frac{{{d_1} - {d_2}}}{{\sqrt {{a^2} + {b^2} + {c^2}} }}} \right|\)

Calculation:

Given: 4x – 2y + 4z + 9 = 0 and 8x – 4y + 8z + 21 = 0

Now, we can rewrite 8x – 4y + 8z + 21 = 0 as 4x – 2y + 4z + (21/2) = 0

So, let P₁ = 4x – 2y + 4z + 9 = 0 and P₂ = 4x – 2y + 4z + (21/2) = 0

As we know that, the distance between two parallel planes ax + by + cz + d₁ = 0 and ax + by + cz + d₂ = 0 is given by: \(\left| {\frac{{{d_1} - {d_2}}}{{\sqrt {{a^2} + {b^2} + {c^2}} }}} \right|\)

Here, a = 4, b = - 2, c = 4, d₁ = 9 and d₂ = 21/2

\(\Rightarrow \;\left| {\frac{{{d_1} - {d_2}}}{{\sqrt {{a^2} + {b^2} + {c^2}} }}} \right| = \left| {\frac{{9 - \frac{{21}}{2}}}{{\sqrt {{4^2} + {{\left( { - \;2} \right)}^2} + {4^2}} }}} \right| = \frac{1}{4}\)