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Three Dimensional Geometry

Short Answer Type

211. Find the equation of the plane through the points (0, – 1, 0), (2, 1, –1) and (1,1,1).

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212. Find the equation of the plane passing through the points (0. – 1, 0), (1, 1, 1) and (3, 3,0).

The equation of any plane through the points (0, – 1, 0) is
a (x – 0) + b (y + 1) + c (z – 0) = 0    ...(1)
∴ it passes through (1, 1, 1)
∴ a + 2 b + c = 0        ...(2)
Again plane passes through (3, 3, 0)
∴ 3 a + 4 b + 0 c = 0    ...(3)
Solve (2) and (3), we get,
   $fraction numerator straight a over denominator 0 minus 4 end fraction space equals fraction numerator straight b over denominator 3 minus 0 end fraction space equals fraction numerator straight c over denominator 4 minus 6 end fraction$

$therefore space space space space space space space fraction numerator straight a over denominator negative 4 end fraction space equals space straight b over 3 space equals space fraction numerator straight c over denominator negative 2 end fraction space space space space space space space space space space space space space space or space space space space space space straight a over 4 space equals space fraction numerator straight b over denominator negative 3 end fraction space equals space straight c over 2 space equals space straight k space space space space space left parenthesis say right parenthesis$

$therefore space space space space space straight a space equals space 4 space straight k comma space space space space straight b space equals space minus 3 space straight k comma space space space space straight c space equals space 2 space straight k$
Putting values of a, b,c in (1), we get,
4 k x – 3 k (y + 1) + 2 k z = 0
or 4 x – 3 y + 2 z = 3
which is required equation of plane.

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Long Answer Type

213. Find the equation of the plane through the three points (1, 1, 1), (1, – 1, 1) and (–7, – 3,–5).

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Short Answer Type

214.

Find the vector equation of the plane passing through the point A(2, 2, –1), B(3, 4, 2) and C (7, 0, 6). Also find the cartesian equation of the plane.

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215. Find the vector equations of the plane passing through the points R(2, 5, –3), S(– 2, – 3, 5) and T(5, 3, – 3).

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216. Find the equations of the planes that passes through three points:
(1, 1,–1), (6, 4, –5),(–4, –2, 3)

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217. Find the equations of the planes that passes through three points:
(1, 1, 0), (1, 2, 1), (–2, 2, –1)

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Long Answer Type

218. If from a point P (a, b, c) perpendiculars PA and PB are drawn to yz and zx-planes, then find the vector equation of the plane OAB.

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219. Find the equation of plane through the points (2, 2, 1), (9, 3, 6), and perpendicular to the plane 2x + 6y + 6z = 9.

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220. Find the equation of the plane through the points (2, 2, 1), (9, 3, 6) and perpendicular to the plane 2x + 6y + 6z – 1 = 0.

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