MHT CET · Maths · Three Dimensional Geometry
The Cartesian equation of the plane, passing through the points \((3,1,1),(1,2,3)\) and \((-1,4,2)\), is
- A \(5 x+6 y-2 z-23=0\)
- B \(-5 x+6 y+2 z+23=0\)
- C \(5 x+6 y+2 z-23=0\)
- D \(5 x-6 y+2 z-23 \doteq 0\)
Answer & Solution
Correct Answer
(C) \(5 x+6 y+2 z-23=0\)
Step-by-step Solution
Detailed explanation
Equation of plane passing through \((3,1,1)\), \((1,2,3)\) and \((-1,4,2)\) is
\(\left|\begin{array}{ccc}x-3 & y-1 & z-1 \\ 1-3 & 2-1 & 3-1 \\ -1-3 & 4-1 & 2-1\end{array}\right|=0\)
\(\Rightarrow\left|\begin{array}{ccc}x-3 & y-1 & z-1 \\ -2 & 1 & 2 \\ -4 & 3 & 1\end{array}\right|=0\)
\(\Rightarrow(x-3)(1-6)-(y-1)(-2+8) +(z-1)\)\((-6+4)=0 \)
\( \Rightarrow-5(x-3)-6(y-1)-2(z-1)=0 \)
\( \Rightarrow-5 x+15-6 y+6-2 z+2=0 \text {. } \)
\( \Rightarrow-5 x-6 y-2 \mathrm{z}+23=0 \)
\( \Rightarrow 5 x+6 y+2 z-23=0\)
\(\left|\begin{array}{ccc}x-3 & y-1 & z-1 \\ 1-3 & 2-1 & 3-1 \\ -1-3 & 4-1 & 2-1\end{array}\right|=0\)
\(\Rightarrow\left|\begin{array}{ccc}x-3 & y-1 & z-1 \\ -2 & 1 & 2 \\ -4 & 3 & 1\end{array}\right|=0\)
\(\Rightarrow(x-3)(1-6)-(y-1)(-2+8) +(z-1)\)\((-6+4)=0 \)
\( \Rightarrow-5(x-3)-6(y-1)-2(z-1)=0 \)
\( \Rightarrow-5 x+15-6 y+6-2 z+2=0 \text {. } \)
\( \Rightarrow-5 x-6 y-2 \mathrm{z}+23=0 \)
\( \Rightarrow 5 x+6 y+2 z-23=0\)
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