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Three Dimensional Geometry

Short Answer Type

241. 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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242. Find the distance of the point (2, 3, 4) from the plane

straight r with rightwards arrow on top. space open parentheses 3 space straight i with hat on top space minus space 6 space straight j with hat on top space plus space 2 space straight k with hat on top close parentheses space equals space minus 11.

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243. Find the distance of a point (2, 5, -3) from the plane

straight r with rightwards arrow on top. space open parentheses 6 straight i with hat on top space minus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top close parentheses space equals space 4.

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244. If the product of distances of the point (1,1,1) from the origin and the plane x – y + z + k = 0 be 5, then find the value of k.

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245. If the points (1, 1, p) and (– 3, 0, 1) be equidistant from the plane $straight r with rightwards arrow on top. space left parenthesis 3 space straight i with hat on top space plus space 4 space straight j with hat on top space minus space 1 space 2 space straight k with hat on top right parenthesis space plus space 13 space equals space 0 comma$ then find value of p.

The equation of the plane is

straight r with rightwards arrow on top. space left parenthesis 3 space straight i with hat on top plus space 4 space straight j with hat on top space minus space 12 space straight k with hat on top right parenthesis space plus 13 space equals 0

open parentheses straight x 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 left parenthesis 3 space straight i with hat on top space plus space 4 space straight j with hat on top space minus space 12 space straight k with hat on top right parenthesis space plus 13 space equals space 0

3 straight x plus 4 straight y minus 12 straight z plus 13 space equals space 0

Since the points (1, 1,p) and (–3, 0, 1) are equidistant from this plane.

therefore space space space space fraction numerator open vertical bar 3 plus 4 minus 12 straight p plus 13 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 9 plus 0 minus 12 plus 13 close vertical bar over denominator square root of 9 plus 16 plus 44 end root end fraction therefore space space space space open vertical bar 20 minus 12 space straight p close vertical bar space equals space open vertical bar negative 8 close vertical bar space space space space space space space space space space rightwards double arrow space space space space open vertical bar 20 space minus space 1 space 2 space straight p close vertical bar space equals space 8 rightwards double arrow space space space space space 20 minus 12 space straight p space equals space plus-or-minus 8 space space space space space space space space space space space space space space rightwards double arrow space space space space space minus 12 space straight p space equals space minus 12 comma space space minus 28 therefore space space space space space space space space space space space space space space straight p space equals space 1 comma space space 7 over 3.

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246.

Find the angle between the line
$straight r with rightwards arrow on top space equals space left parenthesis 2 space straight i with hat on top space plus space 2 space straight j with hat on top space plus space straight k with hat on top right parenthesis space plus space straight lambda space left parenthesis 2 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis$ and the plane
$straight r with rightwards arrow on top. space left parenthesis 3 space straight i with hat on top space minus space 2 space straight j with hat on top space plus space 5 space straight k with hat on top right parenthesis space equals space 4.$

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247.

Find the angle between the line $straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space plus 3 space straight j with hat on top space plus space 9 space straight k with hat on top space plus space straight lambda left parenthesis 2 straight i with hat on top space plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top right parenthesis$ and the plane $straight r with rightwards arrow on top. space open parentheses straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top close parentheses space equals space 5$

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248.

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

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249. Find the angle between the line

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 minus space straight k with hat on top close parentheses space 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

and the plane

straight r with rightwards arrow on top. space open parentheses 2 space 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 equals space 4

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250.

Find the angle between the line $straight x over 4 space equals space fraction numerator straight y minus 2 over denominator 3 end fraction space equals space fraction numerator straight z minus 1 over denominator 6 end fraction$ and the plane x + 2y + 2 + 3 = 0

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Previous Year Papers

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Short Answer Type

245. If the points (1, 1, p) and (– 3, 0, 1) be equidistant from the plane then find value of p.