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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.

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

The equation of line is

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 space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis

therefore space space space space straight b with rightwards arrow on top space equals space straight i with hat on top space space minus space straight j with hat on top space plus space straight k with hat on top

The equation of plane is

straight r with rightwards arrow on top. space space left parenthesis 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 right parenthesis space equals space 4

therefore space space space space straight n 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 straight k with hat on top

Let θ be the angle between the line and the plane

sin space straight theta space equals space fraction numerator straight b with rightwards arrow on top. space straight n with rightwards arrow on top over denominator open vertical bar straight b with rightwards arrow on top close vertical bar space open vertical bar straight n with rightwards arrow on top close vertical bar end fraction space equals space fraction numerator open parentheses straight i with hat on top space minus space straight j with hat on top plus space straight k with hat on top close parentheses 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 over denominator square root of 1 plus 1 plus 1 end root space square root of 4 plus 1 plus 1 end root end fraction space equals space fraction numerator 2 plus 1 plus 1 over denominator square root of 3 space square root of 6 end fraction space equals space fraction numerator 4 over denominator 3 square root of 2 end fraction therefore space space sin space straight theta space equals space fraction numerator 2 square root of 2 over denominator 3 end fraction therefore space space space space straight theta space equals space sin to the power of negative 1 end exponent open parentheses fraction numerator 2 square root of 2 over denominator 3 end fraction close parentheses.

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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

249. Find the angle between the line and the plane