Find the distance of a point (1, –2, 3) from the plane x – y

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 Multiple Choice QuestionsShort Answer Type

271. Find the distance of a point P (5, 3, 4) from the point where the line fraction numerator straight x minus 3 over denominator 1 end fraction space equals fraction numerator straight y minus 4 over denominator 2 end fraction space equals fraction numerator straight z minus 5 over denominator 2 end fraction meets the plane x + y + z = 2.
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272. Find the distance of a point (1, –2, 3) from the plane x – y + z = 5 measured parallel to the line straight x over 2 space equals space straight y over 3 space equals space fraction numerator straight z over denominator negative 6 end fraction


The equation of given plane is
x – y + z = 5    ...(1)
The equation of the line through P (1,–2, 3) parallel to the line straight x over 2 space equals space straight y over 3 space equals fraction numerator straight z over denominator negative 6 end fraction are fraction numerator straight x minus 1 over denominator 2 end fraction space equals space fraction numerator straight y plus 2 over denominator 3 end fraction space equals space fraction numerator straight z minus 3 over denominator negative 6 end fraction

Any point on it is Q (2 r + 1, 3 r –2, – 6 r + 3)
Let it lie on plane (1)
∴    2r + 1 – 3r + 2 – 6r + 3 = 5
therefore space space space space minus 7 space straight r space equals space minus 1 space space space space space space space space rightwards double arrow space space space straight r space equals space 1 over 7
therefore space space space point space straight Q space is space space open parentheses 2 over 7 plus 1 comma space space 3 over 7 minus 2 comma space space fraction numerator negative 6 over denominator 7 end fraction plus 3 close parentheses space space straight i. straight e. space space space open parentheses 9 over 7 comma space minus 11 over 7 comma space 15 over 7 close parentheses

therefore space space space required space distance space PQ space equals space square root of open parentheses 9 over 7 minus 1 close parentheses squared plus open parentheses negative 11 over 7 plus 2 close parentheses squared space plus space open parentheses 15 over 7 minus 3 close parentheses squared end root
space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space equals space square root of 4 over 49 plus 9 over 49 plus 36 over 49 end root space equals space square root of 49 over 49 end root space equals space 1.


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273. Find the distance of the point (- 2, 3, - 4) from the line fraction numerator straight x plus 2 over denominator 3 end fraction space equals space fraction numerator 2 straight y plus 3 over denominator 4 end fraction space equals space fraction numerator 3 straight z plus 4 over denominator 5 end fraction measured  parallel to the plane 4 x + 12 y - 3z + 1 = 0.
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 Multiple Choice QuestionsLong Answer Type

274. Find the distance of the point (–1, –5, –10) from the point of intersection of the line straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space 2 space straight k with hat on top space plus space straight lambda space open parentheses 3 straight i with hat on top space plus space 4 space straight j with hat on top space plus space 2 space straight k with hat on top close parentheses space and space the space plane space straight r with rightwards arrow on top. space open parentheses straight i with hat on top minus straight j with hat on top space plus space space straight k with hat on top close parentheses space equals space 5.
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 Multiple Choice QuestionsShort Answer Type

275. Find the distance between the point with position vector negative straight i with hat on top space minus space 5 space straight j with hat on top space minus space 10 space straight k with hat on top and the point of intersection of the line fraction numerator straight x minus 2 over denominator 3 end fraction space equals space fraction numerator straight y plus 1 over denominator 4 end fraction space equals space fraction numerator straight z minus 2 over denominator 12 end fraction with the plane x – y + z = 5.
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276.

Find the distance of the point (2, 3, 4) from the plane 3x + 2y + 2z + 5 = 0, measured parallel to the line fraction numerator straight x plus 3 over denominator 3 end fraction space equals space fraction numerator straight y minus 2 over denominator 6 end fraction space equals space straight z over 2

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277. Determine whether the lines:
         straight r with rightwards arrow on top space equals space straight i with hat on top space minus space straight j with hat on top space plus space straight lambda open parentheses 2 straight i with hat on top space plus space straight k with hat on top close parentheses
and   straight r with rightwards arrow on top space equals space 2 straight i with hat on top space minus space straight j with hat on top space plus space straight mu space open parentheses 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 space intersect.
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278. Show that the lines
straight r with rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top space plus space straight lambda open parentheses 3 space straight i with hat on top space minus space straight j with hat on top close parentheses space space and space straight r with rightwards arrow on top space equals space 4 space straight i with hat on top space minus space straight k with hat on top space plus space straight mu space open parentheses 2 straight i with hat on top space plus space 3 space straight k with hat on top close parentheses intersect.
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 Multiple Choice QuestionsLong Answer Type

279. Find the equation of plane containing the coplanar lines
straight r with rightwards arrow on top space equals space straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top space plus space straight lambda 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 space and space straight r with rightwards arrow on top space equals space 4 space straight i with hat on top space plus space 2 straight k with hat on top space plus space straight mu open parentheses 2 space 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.
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 Multiple Choice QuestionsShort Answer Type

280. Find the vector and cartesian form of the equation of the plane containing two lines:
            straight r with rightwards arrow on top space equals straight i with hat on top space plus space 2 space straight j with hat on top space minus space 4 space straight k with hat on top space plus space straight lambda left parenthesis 2 straight i with hat on top space plus space 3 space straight j with hat on top space plus space 6 space straight k with hat on top right parenthesis
and       straight r with rightwards arrow on top space equals space 3 straight i with hat on top space plus space 3 space straight j with hat on top space minus space 5 space straight k with hat on top space plus space straight mu space left parenthesis negative 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 8 space straight k with hat on top right parenthesis

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