What is the equation of the plane passing through the point (-2,6,-6),(-3,10,-9) and (-5,0,-6)?

Options

A. `2x-y-2z=2`
B. `2x+y+3z=3`
C. `x+y+z=16`
D. `x-y-z=-3`

Explanation

To find the equation of the plane passing through the points \( A(-2, 6, -6) \), \( B(-3, 10, -9) \), and \( C(-5, 0, -6) \), we can use the determinant method. Here’s a step-by-step solution:

### Step 1: Write the general equation of the plane
The general equation of a plane can be expressed using the determinant of a matrix formed by the coordinates of the points. The equation of the plane passing through the points \( A(x_1, y_1, z_1) \), \( B(x_2, y_2, z_2) \), and \( C(x_3, y_3, z_3) \) is given by:

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

### Step 2: Substitute the points into the determinant
Substituting the coordinates of the points \( A(-2, 6, -6) \), \( B(-3, 10, -9) \), and \( C(-5, 0, -6) \):

- \( (x_1, y_1, z_1) = (-2, 6, -6) \)
- \( (x_2, y_2, z_2) = (-3, 10, -9) \)
- \( (x_3, y_3, z_3) = (-5, 0, -6) \)

The determinant becomes:

\[ \begin{vmatrix} x + 2 & y - 6 & z + 6 \\ -3 + 2 & 10 - 6 & -9 + 6 \\ -5 + 2 & 0 - 6 & -6 + 6 \end{vmatrix} = 0 \]

This simplifies to:

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

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

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

Using the cofactor expansion along the first row:

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

Calculating each of these 2x2 determinants:

1. \( \begin{vmatrix} 4 & -3 \\ -6 & 0 \end{vmatrix} = (4)(0) - (-3)(-6) = 0 - 18 = -18 \)

2. \( \begin{vmatrix} -1 & -3 \\ -3 & 0 \end{vmatrix} = (-1)(0) - (-3)(-3) = 0 - 9 = -9 \)

3. \( \begin{vmatrix} -1 & 4 \\ -3 & -6 \end{vmatrix} = (-1)(-6) - (4)(-3) = 6 + 12 = 18 \)

Putting it all together:

\[ = (x + 2)(-18) - (y - 6)(-9) + (z + 6)(18) \]

Expanding this gives:

\[ -18x - 36 + 9y - 54 + 18z + 108 = 0 \]

### Step 4: Simplify the equation
Combining like terms:

\[ -18x + 9y + 18z + 18 = 0 \]

Dividing through by -9 to simplify:

\[ 2x - y - 2z - 2 = 0 \]

### Final Equation
Thus, the equation of the plane is:

\[ 2x - y - 2z + 2 = 0 \]