MHT CET · Maths · Vector Algebra
If the points \(A(3,2,1), B(4, x, 5), C(4,2,-2)\) and \(D(6,5,-1)\) are coplanar, then \(x\) has the value
- A \(-5\)
- B \(5\)
- C \(-6\)
- D \(6\)
Answer & Solution
Correct Answer
(B) \(5\)
Step-by-step Solution
Detailed explanation
For coplanarity
\(\begin{aligned} & {\left[\begin{array}{ccc}6-3 & 5-2 & -1-1 \\ 4-3 & x-2 & 5-1 \\ 4-3 & 2-2 & -2-1\end{array}\right]=0} \\ & \Rightarrow\left[\begin{array}{ccc}3 & 3 & -2 \\ 1 & x-2 & 4 \\ 1 & 0 & -3\end{array}\right]=0 \end{aligned}\)
\(\Rightarrow 3\{-3(x-2)-0\}-3\{1 \times(-3)-1 \times 4\}-2\)\((1 \times 0-1 \times(x-2))=0 \)
\( \Rightarrow-7 x+35=0 \)
\( \Rightarrow x=5\)
\(\begin{aligned} & {\left[\begin{array}{ccc}6-3 & 5-2 & -1-1 \\ 4-3 & x-2 & 5-1 \\ 4-3 & 2-2 & -2-1\end{array}\right]=0} \\ & \Rightarrow\left[\begin{array}{ccc}3 & 3 & -2 \\ 1 & x-2 & 4 \\ 1 & 0 & -3\end{array}\right]=0 \end{aligned}\)
\(\Rightarrow 3\{-3(x-2)-0\}-3\{1 \times(-3)-1 \times 4\}-2\)\((1 \times 0-1 \times(x-2))=0 \)
\( \Rightarrow-7 x+35=0 \)
\( \Rightarrow x=5\)
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