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Question: Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0 (A). \(\dfrac{5}{2}\...

Distance between two parallel planes 2x + y + 2z = 8 and 4x + 2y + 4z + 5 = 0
(A). 52\dfrac{5}{2}
(B). 72\dfrac{7}{2}
(C). 92\dfrac{9}{2}
(D). 32\dfrac{3}{2}

Explanation

Solution

Before attempting this question, one should have prior knowledge about the concept of planes and also remember to consider the equal of plane i.e. 4x + 2y + 4z + 5 = 0 as equation 2 and divide it by 2, using this information can help you to approach the solution of the question.

Complete step-by-step answer :
According to the given information we have 2 planes parallel to each other represented by the equations 2x + y + 2z = 8 and 4x + 2y + 4z = - 5
Taking 2x + y + 2z = 8 as equation 1 and 4x + 2y + 4z = -5 as equation 2
Now dividing equation 2 by 2 we get
42x+22y+42z=52\dfrac{4}{2}x + \dfrac{2}{2}y + \dfrac{4}{2}z = \dfrac{{ - 5}}{2}
\Rightarrow 2x+y+2z=522x + y + 2z = \dfrac{{ - 5}}{2}
As we know that distance between two parallel planes is given by d=D2D1A2+B2+C2d = \dfrac{{\left| {{D_2} - {D_1}} \right|}}{{\sqrt {{A^2} + {B^2} + {C^2}} }}
Also, we know that A = 2, B = 1 and C = 2 also D1=8{D_1} = 8 and D2=52{D_2} = \dfrac{{ - 5}}{2}
Now substituting the values in the distance formula, we get
Distance between two parallel planes = 8(52)(2)2+(1)2+(2)2\dfrac{{\left| {8 - \left( { - \dfrac{5}{2}} \right)} \right|}}{{\sqrt {{{\left( 2 \right)}^2} + {{\left( 1 \right)}^2} + {{\left( 2 \right)}^2}} }}
\Rightarrow Distance between two parallel planes = 8+524+1+4\dfrac{{\left| {8 + \dfrac{5}{2}} \right|}}{{\sqrt {4 + 1 + 4} }}
\Rightarrow Distance between two parallel planes = 16+529\dfrac{{\left| {\dfrac{{16 + 5}}{2}} \right|}}{{\sqrt 9 }}
\Rightarrow Distance between two parallel planes = 212×3\dfrac{{21}}{{2 \times 3}}
\Rightarrow Distance between two parallel planes = 216=72\dfrac{{21}}{6} = \dfrac{7}{2}
Therefore, the distance between the two parallel planes 72\dfrac{7}{2}
Hence, option B is the correct option.

Note : In the above solution we came across the two terms ``plane” which can be explained as a flat surface which is a two-dimensional surface which is constructed by the combination of two axis such as the combination of x axis and y axis will be x-y plane. Equation of a plane is represented by ax + by + cz = d and this is named as the scalar equation of the plane.