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  4. The equation of the plane passing through the points (1,2,3), (-1, 4, 2) and (3, 1,1) is

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Q. The equation of the plane passing through the points $(1,2,3), (-1, 4, 2)$ and $(3, 1,1)$ is

KEAMKEAM 2018

Solution:

Given that,
$x_{1}=1, y_{1}=2, z_{1}=3$
$x_{2}=-1, y_{2}=4, z_{2}=2$
and $x_{3}=3, y_{3}=1, z_{3}=1$
Equation of plane passing through these points is
$\begin{vmatrix}x-1 & y-2 & z-3 \\ -2 & 2 & -1 \\ 2 & -1 & -2\end{vmatrix}=0$
$\Rightarrow (x-1)(-4-1)-(y-2)(y+2) + (z -3) (2 - 4) =0$
$\Rightarrow (x-1)(-5)-(y-2)(6)+(z-3)(-2)=0$
$\Rightarrow -5 x+5-6 y+12-2 z+6=0$
$\Rightarrow -5 x-6 y-2 z+23=0$
$\Rightarrow 5 x+6 y+2 z-23=0$
$\Rightarrow 5 x+6 y+2 z=23$