Find the vector equation of the plane passing through the inters

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 Multiple Choice QuestionsLong Answer Type

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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).




The equations of given planes are
                    straight r with rightwards arrow on top space. 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 minus 6 space equals space 0                                   ...(1)
and       straight r with rightwards arrow on top. space open parentheses 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 close parentheses space plus space 5 space equals 0                                 ...(2)
Any plane through the intersection of planes (1) and (2) is
                         straight r with rightwards arrow on top. space 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 minus space 6 space plus straight lambda space open curly brackets stack straight r. space with rightwards arrow on top open parentheses 2 space 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 close parentheses space plus space 5 close curly brackets space equals space 0
or              straight r with rightwards arrow on top. space open curly brackets open parentheses 1 plus 2 space straight lambda close parentheses straight i with hat on top space plus space left parenthesis 1 plus 3 space straight lambda right parenthesis space straight j with hat on top space plus space left parenthesis 1 plus 4 space straight lambda right parenthesis space straight k with hat on top space close curly brackets space plus left parenthesis 5 space straight lambda space minus space 6 right parenthesis space equals space 0          ...(3)
It will pass through the point (1, 1, 1)
If     straight r space equals straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top space space satisfies space left parenthesis 3 right parenthesis
straight i. straight e. space space space if space left parenthesis straight i with hat on top plus straight j with hat on top plus straight k with hat on top right parenthesis. space space open curly brackets left parenthesis 1 plus 2 straight lambda right parenthesis space straight i with hat on top space plus space left parenthesis 1 plus 3 space straight lambda right parenthesis space straight j with hat on top space plus space left parenthesis 1 plus 4 space straight lambda right parenthesis space straight k with hat on top close curly brackets space plus space left parenthesis 5 straight lambda minus 6 right parenthesis space equals space 0
straight i. straight e. space if space left parenthesis 1 right parenthesis thin space left parenthesis 1 plus 2 straight lambda right parenthesis plus space left parenthesis 1 right parenthesis thin space left parenthesis 1 plus 3 space straight lambda right parenthesis space plus space left parenthesis 1 right parenthesis thin space left parenthesis 1 plus 4 space straight lambda right parenthesis space plus space left parenthesis 5 space straight lambda space minus space 6 right parenthesis space equals space 0
straight i. straight e. space if space 1 plus 2 straight lambda plus 1 plus 3 straight lambda plus 1 plus 4 straight lambda plus 5 straight lambda minus 6 space equals space 0
straight i. straight e. space if space 14 straight lambda space equals space 3
straight i. straight e. space if space space straight lambda space equals 3 over 14

Putting straight lambda space equals 3 over 14 space in space left parenthesis 3 right parenthesis comma space we space get
              straight r with rightwards arrow on top. space space open curly brackets open parentheses 1 plus 6 over 14 close parentheses straight i with hat on top space plus space open parentheses 1 plus 9 over 14 close parentheses straight j with hat on top space plus space open parentheses 1 plus 12 over 14 close parentheses space straight k with hat on top close curly brackets space plus space open parentheses 15 over 14 minus 6 close parentheses space equals 0
or           straight r with rightwards arrow on top. space left parenthesis 20 space straight i with hat on top space plus space 23 space straight j with hat on top space plus space 26 space space straight k with hat on top right parenthesis minus 69 space equals space 0
or            straight r with rightwards arrow on top. space left parenthesis 20 space straight i with hat on top space plus space 23 space straight j with hat on top space plus space 26 space straight k with hat on top right parenthesis space equals space 69
which is the required vector equation of the plane. 

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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).
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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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 Multiple Choice QuestionsShort 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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 Multiple Choice QuestionsLong 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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 Multiple Choice QuestionsShort Answer Type

210. Find the equation of the plane passing through the points (0, –1, –1), (4, 5, 1) and  (3, 9, 4).
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