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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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155. Find the equation of the plane through the point (3, 4,–1) which is parallel to the plane

straight r with rightwards arrow on top. space open parentheses 2 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.

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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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The equation of plane is
2 x – 3 y + 4 z – 6 = 0 ...(1)
Direction ratios of the normal to the plane (1) are 2, – 3, 4.
Dividing each by

square root of left parenthesis 2 right parenthesis squared plus left parenthesis negative 3 right parenthesis squared plus left parenthesis 4 right parenthesis squared end root space equals space square root of 4 plus 9 plus 16 end root space equals space square root of 29 comma space

we get the direction cosines of the normal as

fraction numerator 2 over denominator square root of 29 end fraction comma space minus fraction numerator 3 over denominator square root of 29 end fraction comma space fraction numerator 4 over denominator square root of 29 end fraction

.
∴ dividing (1) throughout by

square root of 29 comma

we get,

fraction numerator 2 over denominator square root of 29 end fraction straight x plus fraction numerator negative 3 over denominator square root of 29 end fraction straight y space plus space fraction numerator 4 over denominator square root of 29 end fraction straight z space equals space fraction numerator 6 over denominator square root of 29 end fraction

This is of the form lx + my + nz = p, where p is the distance of the plane from the origin.
∴ distance of plane from the origin =

fraction numerator 6 over denominator square root of 29 end fraction.

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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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