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Prove that the image of the point (3,-2,1) in the plane \( 3x-y+4z=2\) lies on the plane \( x+y+z+4=0\).

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  • Mid point of AB where $A(x_1,y_1,z_1)\:and\:B(x_2,y_2,z_2)$ is given by $\bigg(\large\frac{x_1+x_2}{2},\frac{y_1+y_2}{2},\frac{z_1+z_2}{2}\bigg)$
Step 1:
Equation of the given plane is $3x-y+4z=2$----(1)
Let $P(3,-2,1)$ have image as $P'(x_1,y_1,z_1)$
Thus PP' is $\perp$ to the plane (1)
Let $M$ be the mid point of PP' and lie on th plane (1)
Equation of the line PP' is
Any point P' on it is $(3k+3,-k-2,4k+1)$
Since this point lies on the plane
$\therefore k=-\large\frac{1}{2}$
Step 2:
Now substituting the value of k we get the coordinate of M as
$\Rightarrow (\large\frac{3}{2},-\frac{3}{2}$$,-1)$
Since M is the mid point of PP'
$\Rightarrow x_1=0$
$\Rightarrow y_1=-1$
Step 3:
Hence the coordinates of M are (0,-1,-3)
Now substituting for $x,y$ and $z$ in the equation
$\Rightarrow 0-1-3+4=0$
Hence the image of the point $(3,-2,1)$ in the plane $3x-y+4z=2$ lies on the plane.
answered Nov 8, 2013 by sreemathi.v

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