Question

Distance between two parallel planes $2x + y + 2z = 8$ and $4x + 2y + 4z + 5 = 0$ is

$${\text{A}}{\text{. }}\dfrac{3}{2} \\\ {\text{B}}{\text{. }}\dfrac{5}{2} \\\ {\text{C}}{\text{. }}\dfrac{7}{2} \\\ {\text{D}}{\text{. }}\dfrac{9}{2} \\\$$

Answer 1

To find the distance between the two planes when they are parallel to each other. So, assume the two parallel planes $ax + by + cz = {d_1}$ and $ax + by + cz = {d_2}$ then distance between them will be $\dfrac{{\left| {{d_1} - {d_2}} \right|}}{{\sqrt {{a^2} + {b^2} + {c^2}} }}$

Complete step by step answer:
According to the question equations of two parallel planes and they are –
$2x + y + 2z = 8$……..$(i)$
$4x + 2y + 4z + 5 = 0$……..$(ii)$
As we all know that distance between two parallel planes is $\dfrac{{\left| {{d_1} - {d_2}} \right|}}{{\sqrt {{a^2} + {b^2} + {c^2}} }}$, so to make calculation we need to divide equation$(ii)$by 2.
Hence, new equations are: $2x + y + 2z = 8$or $2x + y + 2z - 8 = 0$ and $2x + y + 2z + 5/2 = 0$
We can see that the coefficient of new equations is equal, so it is given:
${d_1} = - 8$ $,$ ${d_2} = \dfrac{5}{2}$ $,a = 2,b = 1,c = 2$
Distance between two parallel planes $= \dfrac{{\left| {{d_1} - {d_2}} \right|}}{{\sqrt {{a^2} + {b^2} + {c^2}} }}$
Putting the value in the formula, we get
$\Rightarrow \dfrac{{\left| { - 8 - \dfrac{5}{2}} \right|}}{{\sqrt {{{(2)}^2} + {{(1)}^2} + {{(2)}^2}} }}$
$\Rightarrow \dfrac{{21}}{6} = \dfrac{7}{2}$

So, the correct answer is “Option C”.

Note: For these types of questions we have to multiply and divide any one of the equations with any number so that the coefficient of the resulting equation will be equal, this makes the calculation easy. Remember the distance formula as it helps to solve the question easily.