What is the equation of the plane passing through point (1,-1,-1) and perpendicular to each of the planes `x-2y-8z=0 and 2x+5y-z=0`?

Options

A. `7x-3y+2z=14`
B. `2x+5y-3z=12`
C. `x-7y+3z=4`
D. `14x-5y+3z=16`

Explanation

To find the equation of the plane that passes through the point (1, -1, -1) and is perpendicular to the given planes \(x - 2y - 8z = 0\) and \(2x + 5y - z = 0\), we can follow these steps:

### Step 1: Find the normal vectors of the given planes
The normal vector of a plane given by the equation \(Ax + By + Cz = D\) is \((A, B, C)\).

- For the plane \(x - 2y - 8z = 0\), the normal vector is:
\[ \mathbf{n_1} = (1, -2, -8) \]

- For the plane \(2x + 5y - z = 0\), the normal vector is:
\[ \mathbf{n_2} = (2, 5, -1) \]

### Step 2: Find the direction vector of the required plane
The required plane is perpendicular to both given planes. Therefore, its normal vector can be found by taking the cross product of \(\mathbf{n_1}\) and \(\mathbf{n_2}\).

\[ \mathbf{n} = \mathbf{n_1} \times \mathbf{n_2} \]

Calculating the cross product:
\[ \mathbf{n} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & -2 & -8 \\ 2 & 5 & -1 \end{vmatrix} \]

Calculating the determinant:
\[ \mathbf{n} = \mathbf{i}((-2)(-1) - (-8)(5)) - \mathbf{j}((1)(-1) - (-8)(2)) + \mathbf{k}((1)(5) - (-2)(2)) \]
\[ = \mathbf{i}(2 + 40) - \mathbf{j}(-1 + 16) + \mathbf{k}(5 + 4) \]
\[ = \mathbf{i}(42) - \mathbf{j}(15) + \mathbf{k}(9) \]
Thus, the normal vector \(\mathbf{n} = (42, -15, 9)\).

### Step 3: Write the equation of the plane
The equation of a plane with normal vector \((A, B, C)\) passing through a point \((x_0, y_0, z_0)\) is given by:
\[ A(x - x_0) + B(y - y_0) + C(z - z_0) = 0 \]

Substituting \(A = 42\), \(B = -15\), \(C = 9\), and the point \((1, -1, -1)\):
\[ 42(x - 1) - 15(y + 1) + 9(z + 1) = 0 \]
Expanding this:
\[ 42x - 42 - 15y - 15 + 9z + 9 = 0 \]
Combining like terms:
\[ 42x - 15y + 9z - 48 = 0 \]
Thus, the equation of the plane is:
\[ 42x - 15y + 9z = 48 \]

### Final Answer
The equation of the plane is:
\[ 42x - 15y + 9z = 48 \]