Find the distance between the planes  and  from Mathematics

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 Multiple Choice QuestionsShort Answer Type

251. Find the angle between the line fraction numerator straight x minus 1 over denominator 1 end fraction space equals space fraction numerator straight y minus 2 over denominator negative 1 end fraction space equals space fraction numerator straight z plus 1 over denominator 1 end fraction and the plane 2x – y + z= 1. 
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252.

Find the angle between the line fraction numerator straight x plus 1 over denominator 3 end fraction space equals space fraction numerator straight y minus 1 over denominator 2 end fraction space equals space fraction numerator straight z minus 2 over denominator 4 end fraction and the plane 2x + y – 3z + 4 = 0.

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253. Find the angle between the lines fraction numerator straight x plus 1 over denominator 2 end fraction space equals space straight y over 3 space equals space fraction numerator straight z minus 3 over denominator 6 end fraction and the plane 10x + 2y – 11z = 3.
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254. Find the angle between the line fraction numerator straight x minus 2 over denominator 3 end fraction space equals space fraction numerator straight y plus 1 over denominator negative 1 end fraction space equals space fraction numerator straight z minus 3 over denominator 2 end fraction and the plane 3x + 4y + z + 5 = 0.
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255.

Find the angle between the line straight r with rightwards arrow on top space equals space open parentheses straight i with hat on top space minus space straight j with hat on top space plus space straight k with hat on top close parentheses space plus space straight lambda space open parentheses 2 straight i with hat on top space minus space straight j with hat on top space plus space 3 space straight k with hat on top close parentheses and the plane straight r with rightwards arrow on top. space open parentheses 2 straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top close parentheses space equals space 4

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256. Find the distance between the parallel planes:
2x – 3y + z + 3 = 0
and 4x – 6y + 2z + 5 = 0
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257. Find the distance between parallel planes:
2x  – y + 3z – 4 = 0 and 6x – 3y + 9z + 13 = 0.
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258. Find the distance between the planes straight r with rightwards arrow on top. space open parentheses straight i with hat on top space plus space 2 space straight j with hat on top space plus space space 3 space straight k with hat on top close parentheses space plus space 7 space equals 0 and straight r with rightwards arrow on top. space open parentheses 2 straight i with hat on top space plus space 4 space straight j with hat on top space plus space 6 space straight k with hat on top close parentheses space plus space 7 space equals space 0.


The equations of the planes are
  straight r with rightwards arrow on top. space open parentheses straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top close parentheses space plus space 7 space equals 0 space space and space straight r with rightwards arrow on top. space left parenthesis 2 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 6 space straight k with hat on top right parenthesis space plus space 7 space equals 0
or             open parentheses straight x stack straight i space with hat on top space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top close parentheses. space space open parentheses straight i with hat on top space plus space 2 space straight j with hat on top space plus space space 3 space straight k with hat on top close parentheses space plus space 7 space equals space 0
and       open parentheses straight x space straight i with hat on top space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top close parentheses space. space open parentheses 2 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space 6 space straight k with hat on top close parentheses space plus space 7 space equals space 0
or space space straight x plus 2 straight y plus 3 straight z plus 7 space equals space 0                                           ...(1)
and 2 straight x plus 4 straight y plus 6 straight z plus 7 space equals space 0                                       ...(2)
These are parallel planes.      
Let (x1, y1 , z1) be any point on plane (1).
∴   x1 + 2y1 + 3z1 + 7 - 0
or x2 + 2y1 + 3z1 = –7    ...(3)
Length of perpendicular from (x1 ,y1 , z1) to the plane (2) is
                   equals space open vertical bar fraction numerator 2 straight x subscript 1 plus space 4 straight y subscript 1 space plus space 6 space straight z subscript 1 plus 7 over denominator square root of 4 plus 16 plus 36 end root end fraction close vertical bar space equals space fraction numerator open vertical bar 2 space left parenthesis straight x subscript 1 space plus space 2 space straight y subscript 1 space plus space 3 space straight z subscript 1 right parenthesis plus 7 close vertical bar over denominator square root of 56 end fraction
space equals space fraction numerator open vertical bar 2 space left parenthesis negative 7 right parenthesis space plus space 7 close vertical bar over denominator square root of 56 end fraction space equals space fraction numerator open vertical bar negative 14 plus 7 close vertical bar over denominator square root of 56 end fraction space equals space fraction numerator open vertical bar negative 7 close vertical bar over denominator square root of 56 end fraction space equals space fraction numerator 7 over denominator square root of 56 end fraction

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 Multiple Choice QuestionsLong Answer Type

259. Find the length and coordinates of the foot of the perpendicular from the point (1, 1, 2) to the plane 2x – 2y + 4z + 5 = 0.
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260. Find the distance between the point P(6, 5, 9) and the plane determined by the points A(3, –1, 2), B(5, 2, 4) and C(– 1, – 1, 6).
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