The value of x when the points `A(2,-1,1),B(4,0,3),C(x,1,1)andD(2,4,3)` are coplanar is

Options

A. 1
B. 0
C. 2
D. `(1)/(2)`

Answer

To find the value of \( x \) when the points \( A(2,-1,1) \), \( B(4,0,3) \), \( C(x,1,1) \), and \( D(2,4,3) \) are coplanar, we can use the concept of vectors and the scalar triple product. The points are coplanar if the scalar triple product of the vectors formed by these points is zero.

### Step-by-Step Solution:

1. **Define the Vectors:**
We will first find the vectors \( \overrightarrow{AB} \), \( \overrightarrow{BC} \), and \( \overrightarrow{CD} \).

- Vector \( \overrightarrow{AB} = B - A = (4 - 2, 0 - (-1), 3 - 1) = (2, 1, 2) \)
- Vector \( \overrightarrow{BC} = C - B = (x - 4, 1 - 0, 1 - 3) = (x - 4, 1, -2) \)
- Vector \( \overrightarrow{CD} = D - C = (2 - x, 4 - 1, 3 - 1) = (2 - x, 3, 2) \)

2. **Set Up the Scalar Triple Product:**
The points are coplanar if the scalar triple product \( \overrightarrow{AB} \cdot (\overrightarrow{BC} \times \overrightarrow{CD}) = 0 \).

We can express this as a determinant:
\[ \begin{vmatrix} 2 & 1 & 2 \\ x - 4 & 1 & -2 \\ 2 - x & 3 & 2 \end{vmatrix} = 0 \]

3. **Calculate the Determinant:**
We will calculate the determinant using the formula for a 3x3 matrix:
\[ \text{Determinant} = a(ei - fh) - b(di - fg) + c(dh - eg) \]
where \( a, b, c \) are the elements of the first row, and \( d, e, f, g, h, i \) are the elements of the second and third rows.

Here, we have:
\[ = 2 \begin{vmatrix} 1 & -2 \\ 3 & 2 \end{vmatrix} - 1 \begin{vmatrix} x - 4 & -2 \\ 2 - x & 2 \end{vmatrix} + 2 \begin{vmatrix} x - 4 & 1 \\ 2 - x & 3 \end{vmatrix} \]

Calculating each of these 2x2 determinants:
- First determinant: \( 1 \cdot 2 - (-2) \cdot 3 = 2 + 6 = 8 \)
- Second determinant: \( (x - 4) \cdot 2 - (-2)(2 - x) = 2x - 8 + 4 - 2x = -4 \)
- Third determinant: \( (x - 4) \cdot 3 - 1(2 - x) = 3x - 12 - 2 + x = 4x - 14 \)

Putting it all together:
\[ 2(8) - 1(-4) + 2(4x - 14) = 0 \]
\[ 16 + 4 + 8x - 28 = 0 \]
\[ 8x - 8 = 0 \]
\[ 8x = 8 \]
\[ x = 1 \]

### Final Answer:
The value of \( x \) when the points are coplanar is \( x = 1 \).