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Find the equation of the plane passing through the intersection of the planes 3x + 2y – z + 1 = 0 and x + y + z – 2 = 0 and the point (2, 2, 1).

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#### Solution

The equation of plane passing through the intersection of the planes 3x + 2y – z + 1 = 0 and x + y + z – 2 = 0 is

`3x + 2y – z + 1)+lambda( x + y + z – 2)=0` ...........(1)

It passes through the point (2, 2, 1)

`(6+4-1+1)+lambda(2+2+1-2)=0`

`10+3lambda=0`

`lambda=-10/3`

Now,

`(3x+2y-z+1)+(-10/3)(x+y+z-2)=0` .......[from(1)]

`9x+6y-3z+3-10x-10y-10z+20=0`

`-x-4y-13z+23=0`

The equation of plane is `x+4y+13z=23`

Concept: Equation of a Plane - Equation of Plane Passing Through the Intersection of Two Given Planes

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