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151. Find equation of a plane parallel to 2x – 3y + z + 9 = 0 and also passing through origin.

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152. Find the equation of the plane through P (1, 4, – 2) that is parallel to the plane – 2 x + y – 3 z = 0.

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153.

Find equation of the plane parallel to x + 3y – 2z + 7 = 0 and passing through the origin.

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154. Find equation of a plane parallel to 3x – 2y + z – 11= 0 and passing through the origin.

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The equation of plane is

straight r with rightwards arrow on top. space open parentheses 2 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 5 space straight k with hat on top close parentheses space plus space 7 space equals space 0

Any plane parallel to it is

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

...(1)
∴ point (3, 4, – 1) lies on it

therefore space space space straight r with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 4 space straight j with hat on top space minus space straight k with hat on top

satisfies (1)

therefore space space space space open parentheses 3 space straight i with hat on top space plus space 4 space straight j with hat on top space minus space straight k with hat on top close parentheses space. space open parentheses 2 space straight i with hat on top space minus space space 3 space straight j with hat on top space plus space 5 space straight k with hat on top close parentheses space equals space straight d rightwards double arrow space space space space left parenthesis 3 right parenthesis thin space left parenthesis 2 right parenthesis space plus space left parenthesis 4 right parenthesis thin space left parenthesis negative 3 right parenthesis space plus space left parenthesis negative 1 right parenthesis thin space left parenthesis 5 right parenthesis space equals space straight d rightwards double arrow space space space space 6 minus 12 minus 5 space equals space straight d space space space space rightwards double arrow space straight d space equals space minus 11

Putting this value of d in (1), we get

straight r with rightwards arrow on top space. space open parentheses 2 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 5 space straight k with hat on top close parentheses space equals space minus 11

straight r with rightwards arrow on top. open parentheses 2 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 5 space straight k with hat on top close parentheses plus 11 space equals space 0 space

which is required equation of plane.

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156. Find the equation of the plane passing through (a, b, c) and parallel to the plane

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157. Find the coordinates of the foot of the perpendicular drawn from the origin to the plane 2x – 3y + 4z – 6 = 0.

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158. Find the distance of the plane 2x – 3 y + 4 z – 6 = 0 from the origin.

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159.

In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
z = 2

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160.

In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
x + y + z = 1

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