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Three Dimensional Geometry

Long Answer Type

221. Find the equation of the plane through the points (1, –1, 2) (2, –2, 2) and perpendicular to the plane 6 x – 2 y + 2 z = 9.

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222. Find the equation of the plane passing through the points (2, 3, – 4), (1, –1, 3) and parallel to x-axis.

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223. Find the equation of the plane passing through the points (3, 4, 1), (0, 1, 0) and parallel to the line

fraction numerator straight x plus 3 over denominator 2 end fraction space equals fraction numerator straight y minus 3 over denominator 7 end fraction space equals space fraction numerator straight z minus 2 over denominator 5 end fraction.

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224. Find the equation of the plane which passes through the points (0, 0, 0) and (3, –1, 2) and is parallel to the line

fraction numerator straight x minus 4 over denominator 1 end fraction space equals space fraction numerator straight y plus 3 over denominator negative 4 end fraction space equals space fraction numerator straight z plus 1 over denominator 7 end fraction.

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225. Find the equation of the plane passing through the points (2, 1, 0), (3, 2, 2) and parallel to the line

fraction numerator straight x minus 1 over denominator 2 end fraction space equals space fraction numerator straight y minus 2 over denominator 3 end fraction space equals space fraction numerator straight z minus 3 over denominator 1 end fraction.

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226. Find the equation of the plane passing through (1, 1, – 1) and perpendicular to planes x + 2 y + 3 z – 7 = 0, 2 x –3 y + 4 z = 0.

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227. Find the equation of the plane passing through the point (–1, –1, 2) and perpendicular to the planes 3x + 2 y – 3 z = 1 and 5 x – 4 y + z = 5.

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228. Find the equation of the plane passing through the point (–1, –1, 2) and perpendicular to the planes 2x + 3 y – 2 z = 5 and x + 2 y – 3 z = 8.

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229. From a point P(1, 2, 4), a perpendicular is drawn on the plane 2x + y – 2z + 3 = 0. Find the equation, the length and coordinates of the foot of the perpendicular.

The equation of plane is
2x + y – 2z + 3 = 0 ...(1)
Direction ratios of the normal to the plane are 2, 1, – 2.
Let M be the foot of perpendicular from P(1, 2, 4) to the plane.
Now PM is a straight line which passes through P(1, 2, 4) and has direction ratios 2, 1,–2.
∴ its equations are

$fraction numerator straight x minus 1 over denominator 2 end fraction space equals space fraction numerator straight y minus 2 over denominator 1 end fraction space equals space fraction numerator straight z minus 4 over denominator negative 2 end fraction space equals space straight r space left parenthesis say right parenthesis$
Any point M on line is. (2 r + 1, r + 2, – 2 r + 4)
∴ M lies on plane (1)
∴ 2 (2 r + 1) + (r + 2) – 2 (–2 r + 4) + 3 = 0
∴ 4 r + 2 + r + 2 + 4 r – 8 + 3 = 0
$therefore space space space space space space space space space space space space space space space space space space space space 9 space straight r space equals space 1 space space space space space space space space space space space space space space space rightwards double arrow space space space space straight r space equals space 1 over 9 therefore space space space space straight M space is space space open parentheses 2 over 9 plus 1 comma space space 1 over 9 plus 2 comma space fraction numerator negative 2 over denominator 9 end fraction plus 4 close parentheses space space space straight i. straight e. space open parentheses 11 over 9 comma space 19 over 9 comma space 34 over 9 close parentheses therefore space space space space foot space of space perpendicular space is space open parentheses 11 over 9 comma space 19 over 9 comma space 34 over 9 close parentheses.$

Length of perpendicular $equals space fraction numerator open vertical bar 2 plus 2 minus 8 plus 3 close vertical bar over denominator square root of 4 plus 1 plus 4 end root end fraction space equals space fraction numerator open vertical bar negative 1 close vertical bar over denominator square root of 9 end fraction space equals space 1 third space units. space$

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230.

Find the coordinates of the image of the point (1, 3, 4) in the plane 2x – y +z + 3 = 0.

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