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Three Dimensional Geometry

Long Answer Type

231.

Find the image of the point (1, 3, 4) in the plane x – y + z = 5.

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Short Answer Type

232. A variable plane passes through a fixed point (a, b, c) and meets the co– ordinate axes in A, B. C. Show that the locus of the point common to the planes through A. B, C parallel to the co-ordinate planes is

straight a over straight x plus straight b over straight y plus straight c over straight z equals 1.

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Long Answer Type

233. Establish by vector method the equation of the plane making intercepts a. b, c on the co-ordinate axes in the form of

straight x over straight a plus straight y over straight b plus straight z over straight c space equals space 1

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Multiple Choice Questions

234. The planes: 2x – y + 4 z = 5 and 5.x – 2.5 y + 10 z = 6 are

Perpendicular
Parallel
intersect y-axis
intersect y-axis

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Short Answer Type

235. Find the distance of the point (3, 4, 5) from the plane

straight r with rightwards arrow on top. space left parenthesis 2 space straight i with hat on top space minus space 5 space straight j with hat on top plus space 3 space straight k with hat on top right parenthesis space equals 13

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236. Find the distance from P (2, 1, – 1 ) to the plane x – 2 y + 4 z = 9.

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237. In the following cases, find the distance of each of the given points from the corresponding given plane.
Point Plane
(0, 0, 0) 3 x - 4 y+12 z = 3

The equation of plane is 3x – 4y+ 12 z – 3 = 0
Let p be length of perpendicular from (0, 0, 0) on plane.

therefore space space space space space space space straight p space equals space fraction numerator open vertical bar 3 space left parenthesis 0 right parenthesis space minus 4 left parenthesis 0 right parenthesis space plus space 12 space left parenthesis 0 right parenthesis space minus space 3 close vertical bar over denominator square root of left parenthesis 3 right parenthesis squared plus left parenthesis negative 4 right parenthesis squared plus left parenthesis 12 right parenthesis squared end root end fraction space equals fraction numerator open vertical bar 0 minus 0 plus 0 minus 3 close vertical bar over denominator square root of 9 plus 16 plus 144 end root end fraction space equals space fraction numerator open vertical bar negative 3 close vertical bar over denominator square root of 169 end fraction space equals 3 over 13

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238. In the following cases, find the distance of each of the given points from the corresponding given plane.
Point Plane
(3, -2, 1) 2x - y + 2z + 3 = 0

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239. In the following cases, find the distance of each of the given points from the corresponding given plane.
Point Plane
(2, 3, -5) x + 2y - 2z = 9

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240. In the following cases, find the distance of each of the given points from the corresponding given plane.
Point Plane
(–6, 0, 0) 2x - 3y + 6z - 2 = 0

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