The equation of the plane through the line of intersection of the planes `ax + by+cz + d= 0` and `a'x + b'y+c'z + d'= 0` parallel to the line `y=0` and `z=0` is

Options

A. `(ab'-a'b)x+(bc'-b'c)y+(ad'-a'd)=0`
B. `(ab'-a'b)x+(bc'-b'c)y+(ad'-a'd)z=0`
C. `(ab'-a'b)y+(bc'-b'c)z+(ad'-a'd)=0`
D. none of these

Explanation

To find the equation of the plane through the line of intersection of the planes \( ax + by + cz + d = 0 \) and \( a'x + b'y + c'z + d' = 0 \), which is parallel to the line defined by \( y = 0 \) and \( z = 0 \), we can follow these steps:

### Step 1: Understand the given planes
We have two planes given by:
1. \( ax + by + cz + d = 0 \) (Plane 1)
2. \( a'x + b'y + c'z + d' = 0 \) (Plane 2)

The line of intersection of these two planes can be represented by a linear combination of their equations.

### Step 2: Write the equation of the desired plane
The equation of the plane through the line of intersection can be expressed as:
\[ a'x + b'y + c'z + d' + \lambda(ax + by + cz + d) = 0 \]
where \( \lambda \) is a parameter.

### Step 3: Simplify the equation
Rearranging gives us:
\[ (a' + \lambda a)x + (b' + \lambda b)y + (c' + \lambda c)z + (d' + \lambda d) = 0 \]
This represents the equation of a plane.

### Step 4: Determine the condition for parallelism
Since the plane is parallel to the line defined by \( y = 0 \) and \( z = 0 \), it must be parallel to the x-axis. This means that the coefficients of \( y \) and \( z \) must be zero:
\[ b' + \lambda b = 0 \quad \text{and} \quad c' + \lambda c = 0 \]

### Step 5: Solve for \( \lambda \)
From the first equation:
\[ \lambda b = -b' \implies \lambda = -\frac{b'}{b} \quad \text{(if } b \neq 0\text{)} \]
From the second equation:
\[ \lambda c = -c' \implies \lambda = -\frac{c'}{c} \quad \text{(if } c \neq 0\text{)} \]

### Step 6: Equate the values of \( \lambda \)
Setting the two expressions for \( \lambda \) equal gives:
\[ -\frac{b'}{b} = -\frac{c'}{c} \]
Cross-multiplying yields:
\[ b'c = b c' \]

### Step 7: Substitute \( \lambda \) back into the plane equation
Substituting \( \lambda = -\frac{b'}{b} \) into the plane equation gives:
\[ \left(a' - \frac{b'}{b} a\right)x + 0 \cdot y + 0 \cdot z + \left(d' - \frac{b'}{b} d\right) = 0 \]
This simplifies to:
\[ \left(a' b - a b'\right)x + \left(d' b - d b'\right) = 0 \]

### Final Equation of the Plane
Thus, the equation of the plane through the line of intersection of the two given planes and parallel to the x-axis is:
\[ (a' b - a b')x + (d' b - d b') = 0 \]