Find the equation of the plane through the points (0, – 1, 0),

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 Multiple Choice QuestionsShort Answer Type

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


The equation of plane passing through (0, – 1, 0) is
A(x – 0) + B(y + 1) + C(z – 0) = 0    ...(1)
∴ it passes through (2, 1, – 1)
∴ A (2 – 0) + B(1 + 1) + C(– 1 – 0) = 0
∴ 2A + 2B – C = 0    ...(2)
Again plane (1) passes through (1, 1, 1)
∴ A( 1 – 0) + B(1 + 1) + C(1 – 0) = 0
∴ A + 2B + C = 0    ....(3)
Solving (2) and (3), we get,
                         fraction numerator straight A over denominator 2 plus 2 end fraction space equals fraction numerator straight B over denominator negative 1 minus 2 end fraction space equals space fraction numerator straight C over denominator 4 minus 2 end fraction

therefore                straight A over 4 space equals space fraction numerator straight B over denominator negative 3 end fraction space equals space straight C over 2 equals straight k space left parenthesis say right parenthesis

therefore space space space space space space space straight A space equals space 4 straight k comma space space space straight B space equals space minus 3 straight k comma space space space straight C space equals 2 space straight k
Putting values of A, B, C in (1), we get,
4 k (x – 0) – 3 k (y + 1) + 2 k (z – 0) = 0
∴ 4 x – 3 y + 2 z = 0
∴ 4 x – 3 y + 2 z = 3
which is required equation of plane.

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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).
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 Multiple Choice QuestionsLong 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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 Multiple Choice QuestionsShort 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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 Multiple Choice QuestionsLong 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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