Correct Answer - Option 3 : 6x - 4z + 6 = 0
Concept:
Cartesian equation of plane passing through the three non-collinear points (x1, y1, z1), (x2, y2, z2) and (x3, y3, z3) is given by: \(\begin{vmatrix} x-x_{1} & y-y_{1} &z-z_{1} \\ x_{2}-x_{1} &y_{2}-y_{1} &z_{2}-z_{1} \\ x_{3}-x_{1} &y_{3}-y_{1} & z_{3}-z_{1} \end{vmatrix}=0\)
Calculation:
cartesian equation of plane passing through the non-collinear points (1, 2, 3), (1, 0, 3) and (-1, 2, 0) is given by: \(\begin{vmatrix} x-1 & y-2 &z-3 \\ 1-1 &0-2 &3-3 \\ -1-1 &2-2 & 0-3 \end{vmatrix}=0\)
⇒ \(\begin{vmatrix} x-1 & y-2 &z-3 \\ 0 &-2 &0 \\ -2 &0 & -3 \end{vmatrix}=0\)
⇒ (x - 1)(6 - 0) - (y - 2)(0 - 0) + (z - 3)(0 - 4) = 0.
⇒ 6(x - 1) - 0 + (-4)(z - 3) = 0
⇒ 6x - 6 - 4z + 12 = 0
⇒ 6x - 4z + 6 = 0
Hence, option 3 is correct.
