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

Let the equation of plane be

straight r with rightwards arrow on top. space straight n with rightwards arrow on top space equals space straight d

...(1)
Let the plane meet the axes in A, B, C respectively such that OA = a, OB = b, OC = c
A with position

straight a space straight i with rightwards arrow on top

lies on the plane (1)

therefore space space space space straight a space space straight i with hat on top space. space straight n with hat on top space space equals space straight d space space space space space space space rightwards double arrow space space space space straight i with rightwards arrow on top. space straight n with rightwards arrow on top space equals space straight d over straight a

...(2)
Similarly,

straight j with rightwards arrow on top. space straight n with rightwards arrow on top space equals space straight d over straight b

...(3)

and

straight k with rightwards arrow on top. space straight n with rightwards arrow on top space equals space straight d over straight c

...(4)
Now (1) can be written as

open parentheses straight x space straight i with rightwards arrow on top space plus space straight y space straight j with rightwards arrow on top space plus space straight z space straight k with rightwards arrow on top close parentheses space. space straight n with rightwards arrow on top space equals space straight d

or space space space straight x space straight i with rightwards arrow on top space. space straight n with rightwards arrow on top space plus space straight y space straight j with rightwards arrow on top space. space straight n with rightwards arrow on top space plus space straight z space straight k with rightwards arrow on top space. space straight n with rightwards arrow on top space equals space straight d or space space straight x space open parentheses straight d over straight a close parentheses space plus space straight y open parentheses straight d over straight b close parentheses space plus space straight z space open parentheses straight d over straight c close parentheses space equals space straight d or space space straight x over straight a plus straight y over straight b plus straight z over straight c space equals space 1

which is required equation of plane x.

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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

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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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