The straight line through the point of intersection of `ax + by+c=0` and `a'x+b'y+c'= 0` are parallel to the y-axis has the equation

Options

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

Explanation

To find the equation of the straight line that passes through the point of intersection of the lines \( ax + by + c = 0 \) and \( a'x + b'y + c' = 0 \) and 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 of the lines \( ax + by + c = 0 \) and \( a'x + b'y + c' = 0 \), we can solve these two equations simultaneously.

1. Rearranging the first equation:
\[ by = -ax - c \quad \Rightarrow \quad y = -\frac{a}{b}x - \frac{c}{b} \]

2. Substitute \( y \) from the first equation into the second equation:
\[ a'x + b'\left(-\frac{a}{b}x - \frac{c}{b}\right) + c' = 0 \]
Simplifying this will give us the x-coordinate of the intersection point.

### Step 2: Solve for the x-coordinate
Rearranging the equation:
\[ a'x - \frac{b'a}{b}x - \frac{b'c}{b} + c' = 0 \]
Combine like terms:
\[ \left(a' - \frac{b'a}{b}\right)x + \left(c' - \frac{b'c}{b}\right) = 0 \]
From this, we can solve for \( x \):
\[ x = \frac{\frac{b'c}{b} - c'}{a' - \frac{b'a}{b}} \quad \text{(if the denominator is not zero)} \]

### Step 3: Find the y-coordinate
Substituting the value of \( x \) back into either of the original equations will yield the corresponding \( y \)-coordinate.

### Step 4: Equation of the line parallel to the y-axis
A line parallel to the y-axis has the form \( x = k \), where \( k \) is the x-coordinate of the point of intersection. Therefore, the equation of the required line can be expressed as:
\[ x = \text{(x-coordinate of intersection)} \]

### Step 5: General form of the equation
The general form of the line passing through the point of intersection and parallel to the y-axis can be expressed as:
\[ x \cdot b' - a' \cdot b + c \cdot b' - c' \cdot b = 0 \]
This represents the required line in the desired format.

### Final Answer
Thus, the equation of the straight line through the point of intersection of the two given lines and parallel to the y-axis is:
\[ x \cdot b' - a' \cdot b + c \cdot b' - c' \cdot b = 0 \]

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