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

The Cartesian equation of the plane, passing through the points $(3,1,1),(1,2,3)$ and $(-1,4,2)$, is
  1. $5 x+6 y-2 z-23=0$
  2. $-5 x+6 y+2 z+23=0$
  3. $5 x+6 y+2 z-23=0$
  4. $5 x-6 y+2 z-23 \doteq 0$

Solution

Equation of plane passing through $(3,1,1)$, $(1,2,3)$ and $(-1,4,2)$ is $\begin{aligned} & \left|\begin{array}{ccc} x-3 & y-1 & z-1 \\ 1-3 & 2-1 & 3-1 \\ -1-3 & 4-1 & 2-1 \end{array}\right|=0 \\ & \Rightarrow\left|\begin{array}{ccc} x-3 & y-1 & z-1 \\ -2 & 1 & 2 \\ -4 & 3 & 1 \end{array}\right|=0 \\ & \Rightarrow(x-3)(1-6)-(y-1)(-2+8) +(z-1)(-6+4)=0 \\ & \Rightarrow-5(x-3)-6(y-1)-2(z-1)=0 \\ & \Rightarrow-5 x+15-6 y+6-2 z+2=0 \text {. } \\ & \Rightarrow-5 x-6 y-2 \mathrm{z}+23=0 \\ & \Rightarrow 5 x+6 y+2 z-23=0 \end{aligned}$

Asked in: MHT CET 2024 (02 May Shift 1)

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