The line through the pont of intersection of lines `ax+by+c=0` and `dx+b'y+c'=0` which is parallel to y-axis is

Options

A. `x(ab'-d'b)+(cb'-c'b)=0`
B. `x(ab'-a'b)+(cb'+c'b)=0`
C. `y(ab'-a'b)+(ac'-d'c)=0`
D. none

Answer

To find the equation of the line through the point of intersection of the lines \( ax + by + c = 0 \) and \( dx + b'y + c' = 0 \) that is parallel to the y-axis, we can follow these steps:

### Step 1: Find the point of intersection of the two lines
To find the point of intersection, we need to solve the system of equations given by the two lines.

1. The first line is \( ax + by + c = 0 \).
2. The second line is \( dx + b'y + c' = 0 \).

We can solve these equations simultaneously to find the coordinates \( (x_0, y_0) \) of the intersection point.

### Step 2: Write the equation of the line through the intersection point
The general form of the equation of a line through a point \( (x_0, y_0) \) can be expressed as:

\[ L_1 + \lambda L_2 = 0 \]

Where \( L_1 \) is the equation of the first line and \( L_2 \) is the equation of the second line.

Substituting the equations of the lines, we have:

\[ (ax + by + c) + \lambda (dx + b'y + c') = 0 \]

### Step 3: Rearranging the equation
Rearranging the above equation gives:

\[ (ax + \lambda dx) + (by + \lambda b'y) + (c + \lambda c') = 0 \]

This can be grouped as:

\[ (a + \lambda d)x + (b + \lambda b')y + (c + \lambda c') = 0 \]

### Step 4: Condition for the line to be parallel to the y-axis
For the line to be parallel to the y-axis, the coefficient of \( y \) must be zero. Thus, we set:

\[ b + \lambda b' = 0 \]

Solving for \( \lambda \):

\[ \lambda = -\frac{b}{b'} \]

### Step 5: Substitute \( \lambda \) back into the equation
Substituting \( \lambda \) back into the equation gives:

\[ (a - \frac{b}{b'}d)x + 0 \cdot y + (c - \frac{b}{b'}c') = 0 \]

This simplifies to:

\[ (a - \frac{bd}{b'})x + (c - \frac{bc'}{b'}) = 0 \]

### Step 6: Final equation of the line
Rearranging gives us the final equation of the line:

\[ (a - \frac{bd}{b'})x + (c - \frac{bc'}{b'}) = 0 \]

### Conclusion
Thus, the equation of the line through the point of intersection of the given lines that is parallel to the y-axis is:

\[ (a - \frac{bd}{b'})x + (c - \frac{bc'}{b'}) = 0 \]