MHT CET · Maths · Vector Algebra
If the points \(\mathrm{A}(2,1,-1), \mathrm{B}(0,-1,0), \mathrm{C}(4,0,4)\) and \(\mathrm{D}(2,0, x)\) are coplanar, then \(x=\)
- A 2
- B 1
- C 4
- D 3
Answer & Solution
Correct Answer
(B) 1
Step-by-step Solution
Detailed explanation
(B)
\(\begin{aligned}
\overline{\mathrm{AB}} &=-2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}} \\
\overline{\mathrm{AC}} &=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+5 \hat{\mathrm{k}} \\
\overline{\mathrm{AD}} &=-\hat{\mathrm{j}}+(\mathrm{x}+1) \hat{\mathrm{k}}
\end{aligned}\)
Since the points A, B, C, D are coplanar, we write
\(\begin{aligned}
&\left|\begin{array}{ccc}
-2 & -2 & 1 \\
2 & -1 & 5 \\
0 & -1 & x+1
\end{array}\right|=0\end{aligned}\)
\(\therefore -2[(-1-x)-(-5)]+2[(2 x+2)-0]+1[-2-0]=0\)
\(-2[-1-x+5]+2[2 x+2]+[-2]=0 \Rightarrow-8+2 x\) \(+~4 x+2=0\Rightarrow x=1\)
\(\begin{aligned}
\overline{\mathrm{AB}} &=-2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}} \\
\overline{\mathrm{AC}} &=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+5 \hat{\mathrm{k}} \\
\overline{\mathrm{AD}} &=-\hat{\mathrm{j}}+(\mathrm{x}+1) \hat{\mathrm{k}}
\end{aligned}\)
Since the points A, B, C, D are coplanar, we write
\(\begin{aligned}
&\left|\begin{array}{ccc}
-2 & -2 & 1 \\
2 & -1 & 5 \\
0 & -1 & x+1
\end{array}\right|=0\end{aligned}\)
\(\therefore -2[(-1-x)-(-5)]+2[(2 x+2)-0]+1[-2-0]=0\)
\(-2[-1-x+5]+2[2 x+2]+[-2]=0 \Rightarrow-8+2 x\) \(+~4 x+2=0\Rightarrow x=1\)
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