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…

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 $x=$
  1. 2
  2. 1
  3. 4
  4. 3

Solution

$\begin{aligned} \overline{\mathrm{AB}} &=-2 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+\hat{\mathrm{k}} \\ \overline{\mathrm{AC}} &=2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+5 \hat{\mathrm{k}} \\ \overline{\mathrm{AD}} &=-\hat{\mathrm{j}}+(\mathrm{x}+1) \hat{\mathrm{k}} \end{aligned}$ Since the points A, B, C, D are coplanar, we write $\begin{aligned} &\left|\begin{array}{ccc} -2 & -2 & 1 \\ 2 & -1 & 5 \\ 0 & -1 & x+1 \end{array}\right|=0 \\ \therefore &-2[(-1-x)-(-5)]+2[(2 x+2)-0]+1[-2-0]=0 \\ &-2[-1-x+5]+2[2 x+2]+[-2]=0 \Rightarrow-8+2 x+4 x+2=0 \quad \Rightarrow x=1 \end{aligned}$

Asked in: MHT CET 2020 (19 Oct Shift 1)

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