Let the plane `2x - 3y + 9z = 0` is P, the equation of line passing through `(0,1 , -1)` and lying-in plane `4x - 2y + 7x + 9 = 0` & parallel to P is:

Options

A. `(x)/(-3) = (y - 1)/2 = (z + 1)/8`
B. `(x)/(-3) = (y - 1)/(22) = (z + 1)/(8)`
C. `(x - 1)/(-3) = (y - 1)/(2) = (z + 1)/(8)`
D. `(x)/(-3) = (y - 1)/(11) = (z + 1)/(4)`

Explanation

To find the equation of the line that passes through the point \((0, 1, -1)\), lies in the plane \(4x - 2y + 7z + 9 = 0\), and is parallel to the plane \(2x - 3y + 9z = 0\), we can follow these steps:

### Step 1: Identify the normal vectors of the planes
The normal vector of the plane \(P: 2x - 3y + 9z = 0\) can be identified from its coefficients:
- Normal vector \(n_1 = (2, -3, 9)\)

For the plane \(4x - 2y + 7z + 9 = 0\):
- Normal vector \(n_2 = (4, -2, 7)\)

### Step 2: Find the direction vector of the line
To find a direction vector for the line that is parallel to the plane \(P\) and lies in the plane defined by \(n_2\), we need to compute the cross product of the two normal vectors \(n_1\) and \(n_2\).

\[ \text{Direction vector } b = n_1 \times n_2 \]

Calculating the cross product:

\[ b = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & -3 & 9 \\ 4 & -2 & 7 \end{vmatrix} \]

Calculating the determinant:

\[ b = \hat{i}((-3)(7) - (9)(-2)) - \hat{j}((2)(7) - (9)(4)) + \hat{k}((2)(-2) - (-3)(4)) \]

Calculating each component:

- \( \hat{i}: -21 + 18 = -3 \)
- \( \hat{j}: 14 - 36 = -22 \) (note the negative sign in front)
- \( \hat{k}: -4 + 12 = 8 \)

Thus, the direction vector \(b = (-3, 22, 8)\).

### Step 3: Write the equation of the line
The equation of a line in 3D can be expressed as:

\[ \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} \]

Where \((x_1, y_1, z_1)\) is a point on the line and \((a, b, c)\) are the components of the direction vector. Here, \((x_1, y_1, z_1) = (0, 1, -1)\) and \((a, b, c) = (-3, 22, 8)\).

Substituting these values into the equation gives:

\[ \frac{x - 0}{-3} = \frac{y - 1}{22} = \frac{z + 1}{8} \]

### Final Answer
The equation of the line is:

\[ \frac{x}{-3} = \frac{y - 1}{22} = \frac{z + 1}{8} \]