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

Answer

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

### Step 1: Understand the Given Planes
The two planes are given by:
1. \( E_1: ax + by + cz + d = 0 \)
2. \( E_2: a'x + b'y + c'z + d' = 0 \)

### Step 2: Find the Equation of the Plane Through the Line of Intersection
The equation of the plane through the line of intersection of two planes can be expressed as:
\[ E_2 + \lambda E_1 = 0 \]
This means:
\[ a'x + b'y + c'z + d' + \lambda(ax + by + cz + d) = 0 \]
Expanding this gives:
\[ (a' + \lambda a)x + (b' + \lambda b)y + (c' + \lambda c)z + (d' + \lambda d) = 0 \]

### Step 3: Apply the Condition for Parallelism
Since the required plane is parallel to the line \( y = 0 \) and \( z = 0 \), it implies that the plane must be vertical and can be expressed in the form:
\[ A x + B = 0 \]
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 4: Solve for \( \lambda \)
From \( b' + \lambda b = 0 \):
\[ \lambda = -\frac{b'}{b} \quad (b \neq 0) \]
From \( c' + \lambda c = 0 \):
\[ \lambda = -\frac{c'}{c} \quad (c \neq 0) \]
Equating the two expressions for \( \lambda \):
\[ -\frac{b'}{b} = -\frac{c'}{c} \]
This gives us the relationship:
\[ \frac{b'}{b} = \frac{c'}{c} \]

### Step 5: Substitute \( \lambda \) Back into the Plane Equation
Substituting \( \lambda = -\frac{b'}{b} \) into the plane equation:
\[ (a' - \frac{b'}{b} a)x + 0 \cdot y + 0 \cdot z + (d' - \frac{b'}{b} d) = 0 \]
This simplifies to:
\[ (a' - \frac{b'}{b} a)x + (d' - \frac{b'}{b} d) = 0 \]

### Step 6: Final Equation of the Plane
The final equation of the required plane can be expressed as:
\[ (a' - \frac{b'}{b} a)x + (d' - \frac{b'}{b} d) = 0 \]
This is the equation of the plane through the line of intersection of the given planes and parallel to the specified line.