The four points `(0,4,3)(-1,-5,-3),`
`(-2-2,1) and (1,1,-1)` lie in the plane

Options

A. `4x+3y+2z-9=0`
B. `9x-5y+6z+2=0`
C. `3x+4y+7z-5=0`
D. None

Answer

To find the equation of the plane that passes through the points \( A(0, 4, 3) \), \( B(-1, -5, -3) \), and \( C(-2, -2, 1) \), we can use the determinant form of the equation of a plane. Here’s the step-by-step solution:

### Step 1: Set up the determinant form
The equation of a plane through three points \( A(x_1, y_1, z_1) \), \( B(x_2, y_2, z_2) \), and \( C(x_3, y_3, z_3) \) can be expressed using the determinant:

\[ \begin{vmatrix} x - x_1 & y - y_1 & z - z_1 \\ x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ x_3 - x_1 & y_3 - y_1 & z_3 - z_1 \end{vmatrix} = 0 \]

### Step 2: Substitute the points into the determinant
Using the points \( A(0, 4, 3) \), \( B(-1, -5, -3) \), and \( C(-2, -2, 1) \):

- \( x_1 = 0, y_1 = 4, z_1 = 3 \)
- \( x_2 = -1, y_2 = -5, z_2 = -3 \)
- \( x_3 = -2, y_3 = -2, z_3 = 1 \)

The determinant becomes:

\[ \begin{vmatrix} x - 0 & y - 4 & z - 3 \\ -1 - 0 & -5 - 4 & -3 - 3 \\ -2 - 0 & -2 - 4 & 1 - 3 \end{vmatrix} = 0 \]

This simplifies to:

\[ \begin{vmatrix} x & y - 4 & z - 3 \\ -1 & -9 & -6 \\ -2 & -6 & -2 \end{vmatrix} = 0 \]

### Step 3: Calculate the determinant
Now, we calculate the determinant:

\[ \begin{vmatrix} x & y - 4 & z - 3 \\ -1 & -9 & -6 \\ -2 & -6 & -2 \end{vmatrix} \]

Using the formula for a 3x3 determinant:

\[ = x \begin{vmatrix} -9 & -6 \\ -6 & -2 \end{vmatrix} - (y - 4) \begin{vmatrix} -1 & -6 \\ -2 & -2 \end{vmatrix} + (z - 3) \begin{vmatrix} -1 & -9 \\ -2 & -6 \end{vmatrix} \]

Calculating each of the 2x2 determinants:

1. \( \begin{vmatrix} -9 & -6 \\ -6 & -2 \end{vmatrix} = (-9)(-2) - (-6)(-6) = 18 - 36 = -18 \)

2. \( \begin{vmatrix} -1 & -6 \\ -2 & -2 \end{vmatrix} = (-1)(-2) - (-6)(-2) = 2 - 12 = -10 \)

3. \( \begin{vmatrix} -1 & -9 \\ -2 & -6 \end{vmatrix} = (-1)(-6) - (-9)(-2) = 6 - 18 = -12 \)

Substituting back into the determinant:

\[ = x(-18) - (y - 4)(-10) + (z - 3)(-12) \]

This simplifies to:

\[ -18x + 10(y - 4) - 12(z - 3) = 0 \]

### Step 4: Expand and simplify
Expanding this gives:

\[ -18x + 10y - 40 - 12z + 36 = 0 \]

Combining like terms:

\[ -18x + 10y - 12z - 4 = 0 \]

### Step 5: Rearranging the equation
To express the equation in a standard form, we can multiply through by -1:

\[ 18x - 10y + 12z + 4 = 0 \]

### Step 6: Final equation of the plane
Thus, the equation of the plane is:

\[ 9x - 5y + 6z + 2 = 0 \]