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Three Dimensional Geometry

Long Answer Type

201. Find the vector equation of the plane passing through the intersection of the planes:

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 6 comma space space space space straight r with rightwards arrow on top. space space open parentheses 2 space straight i with hat on top space plus space space 3 space straight j with hat on top space plus space 4 space straight k with hat on top close parentheses space equals space minus 5

and the point (1, 1, 1).

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202. Find the vector equation of the plane passing through the intersection of the planes $straight r with rightwards arrow on top. space open parentheses 2 straight i with hat on top space plus space straight j with hat on top space plus space 3 space straight k with hat on top close parentheses space equals 7 comma space space space space space straight r with rightwards arrow on top space. space open parentheses 2 space straight i with hat on top space plus space 5 space straight j with hat on top space plus space 3 space straight k with hat on top close parentheses space equals space 9$ and the point (2, 1, 3).

The equations of given planes are

straight r with rightwards arrow on top. space left parenthesis 2 space straight i with hat on top space plus space straight j with hat on top space plus space space 3 space straight k with hat on top right parenthesis space equals space 7

i.e.

left parenthesis straight x space 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 right parenthesis. space space left parenthesis 2 space straight i with hat on top space plus space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis space equals space 7

i.e.

2 straight x plus straight y plus 3 straight z minus 7 space equals space 0

...(1)
and

straight r with rightwards arrow on top. space left parenthesis 2 straight i with hat on top space plus space 5 space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis space equals space 9

i.e.

left parenthesis straight x space 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 right parenthesis. space space space open parentheses 2 space straight i with hat on top space plus space straight j with hat on top space plus space 3 space straight k with hat on top close parentheses space equals space 9

i.e.

2 straight x plus 5 straight y plus 3 straight z minus 9 space equals space 0

...(2)
Any plane through the intersection of (1) and (2) is
(2x + y + 3z – 7) + k (2x + 5y + 3z – 9) = 0 ...(3)
∴ it passes through (2, 1, 3)

therefore space space space left parenthesis 4 plus 1 plus 9 minus 7 right parenthesis space plus space straight k left parenthesis 4 plus 5 plus 9 minus 9 right parenthesis equals space 0 therefore space space space space 7 plus 9 space straight k space equals space 0 space space space space space space space space space rightwards double arrow space space space space space straight k space equals space minus 7 over 9

Putting

straight k space equals space minus 7 over 9

in (3), we get,

left parenthesis 2 straight x plus straight y plus 3 straight z minus 7 right parenthesis space minus 7 over 9 left parenthesis 2 straight x plus 5 straight y plus 3 straight z minus 9 right parenthesis space equals space 0

or space space space 9 space left parenthesis 2 straight x plus straight y plus 3 straight z minus 7 right parenthesis space minus space 7 space left parenthesis 2 straight x plus 5 straight y plus 3 straight z minus 9 right parenthesis space equals space 0 or space space 18 straight x plus 9 straight y plus 27 straight z minus 63 minus 14 straight x minus 35 straight y minus 21 straight z plus 63 space equals 0 or space space 4 straight x minus 26 straight y plus 6 straight z space equals space 0 space space space space space space space or space space space 2 straight x minus 13 straight y plus 3 straight z space equals space 0 or space space space space space left parenthesis straight x straight i with hat on top space plus space straight y straight j with hat on top space plus space straight z space straight k with hat on top right parenthesis. space space left parenthesis 2 space straight i with hat on top space minus space 13 space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis space equals space 0 or space space space straight r with rightwards arrow on top. space left parenthesis 2 straight i with hat on top space minus space 13 space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis space equals space 0

which is vector equation of required plane.

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203. Find the equation of the plane passing through the line of intersection of the planes

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 1 space space space and space space straight r with rightwards arrow on top. space space open parentheses 2 space straight i with hat on top space plus space 3 space straight j with hat on top space minus space straight k with hat on top close parentheses space plus space 4 space space equals 0

and parallel to x-axis.

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204. Find the equation of the plane through the line of intersection of the planes 3x – 4y + 5z = 10, 2x + 2y – 3z = 4 and parallel to the line x = 2y = 3z.

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205. Find the equation of the plane which contains the line of intersection of the planes

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 plus space 3 space straight k with hat on top close parentheses space minus space 4 space equals 0 space space space and space space straight r with rightwards arrow on top. space space left parenthesis 2 straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top right parenthesis space plus space 5 space equals space 0

and which is perpendicular to the plane

straight r with rightwards arrow on top. space open parentheses 5 straight i with hat on top space plus space 3 straight j with hat on top space minus space 6 straight k with hat on top space close parentheses space plus space 8 space equals 0 space.

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Short Answer Type

206. Find the equation of plane passing through the line of intersection of the planes 2x – y = 0 and 3z – y = 0 and perpendicular to the plane 4x + 5y – 3Z = 8.

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207. Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0.

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Long Answer Type

208. Find the equation of the plane which is perpendicular to the plane 5x + 3y + 6z + 8 = 0 and which contains the line of intersection of the planes x + 2y + 3z – 4 = 0 and 2x + y – z + 5 = 0.

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209. Find the equation of the plane passing through the line of intersection of planes x + y – 2 z + 3 = 0 and 3 x – y – 2 z – 4 = 0 and perpendicular to the plane 2x + 3 y – z + 1 = 0.

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