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Three Dimensional Geometry

Question
CBSEENMA12033486
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Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8 z = 12 is

  • 2 units

  • 4 units

  • 8 units

  • fraction numerator 2 over denominator square root of 29 end fraction units

Solution

D.

fraction numerator 2 over denominator square root of 29 end fraction units The equations of given planes
2x + 3y + 4z - 4 = 0        ...(1)
and 4x + 6y +8 z – 12 = 0    ...(2)
Here 2 over 4 space equals space 3 over 6 space equals 4 over 8
∴ planes (1) and (2) are parallel.
Let (x1,y1.,z1) be any point on plane (1).
∴ 2x1 + 3y1 + 4 z1 – 4 = 0
or  2x + 31+ 4z1 = 4    ...(3)
Length of perpendicular from (x1, y1, z1,) to the plane (2) is
equals space fraction numerator open vertical bar 4 straight x subscript 1 space plus space 6 straight y subscript 1 plus 8 straight z subscript 1 minus 12 close vertical bar over denominator square root of 16 plus 36 plus 64 end root end fraction space equals space fraction numerator open vertical bar 2 space left parenthesis 2 straight x subscript 1 space plus space 3 straight y subscript 1 plus space 4 space straight z subscript 1 right parenthesis space minus space 12 close vertical bar over denominator square root of 116 end fraction equals space fraction numerator open vertical bar 2 space left parenthesis 4 right parenthesis space minus space 12 close vertical bar over denominator square root of 116 end fraction space equals space fraction numerator open vertical bar 8 minus 12 close vertical bar over denominator square root of 116 end fraction space equals space fraction numerator open vertical bar negative 4 close vertical bar over denominator square root of 4 space cross times 29 end root end fraction space space space space space space space space space space space space space space space space space space space space space space space space open square brackets because space space of space left parenthesis 3 right parenthesis close square brackets equals space fraction numerator 4 over denominator 2 square root of 29 end fraction space equals space fraction numerator 2 over denominator square root of 29 end fraction therefore space space space space left parenthesis straight D right parenthesis space is space correct space answer.

Some More Questions From Three Dimensional Geometry Chapter

If α, β, γ are the angles which a line makes with the axes, prove that sin2α + sin2 β + sin2γ = 2.

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