The equations `ax+by+c=0` and `dx+ey+f=0` represent the same straight line if and only if

Options

A. `(a)/(d)=(b)/(c)`
B. `c=f`
C. `(a)/(d)=(b)/(e)=(c)/(f)`
D. `a=d,b=e,c=f`

Answer

To determine the condition under which the equations \( ax + by + c = 0 \) and \( dx + ey + f = 0 \) represent the same straight line, we can follow these steps:

### Step 1: Identify the slopes of the lines
The slope of a line in the form \( Ax + By + C = 0 \) is given by the formula:
\[ \text{slope} = -\frac{A}{B} \]
For the first line \( ax + by + c = 0 \):
\[ \text{slope}_1 = -\frac{a}{b} \]
For the second line \( dx + ey + f = 0 \):
\[ \text{slope}_2 = -\frac{d}{e} \]

### Step 2: Set the slopes equal
For the two lines to be the same, their slopes must be equal:
\[ -\frac{a}{b} = -\frac{d}{e} \]
This simplifies to:
\[ \frac{a}{d} = \frac{b}{e} \]

### Step 3: Find the y-intercepts
To find the y-intercept of the first line, we can rearrange the equation:
\[ by = -ax - c \implies y = -\frac{a}{b}x - \frac{c}{b} \]
Thus, the y-intercept \( y_1 \) is:
\[ y_1 = -\frac{c}{b} \]

For the second line:
\[ ey = -dx - f \implies y = -\frac{d}{e}x - \frac{f}{e} \]
Thus, the y-intercept \( y_2 \) is:
\[ y_2 = -\frac{f}{e} \]

### Step 4: Set the y-intercepts equal
For the two lines to be the same, their y-intercepts must also be equal:
\[ -\frac{c}{b} = -\frac{f}{e} \]
This simplifies to:
\[ \frac{c}{b} = \frac{f}{e} \]

### Step 5: Combine the conditions
From the previous steps, we have two conditions:
1. \( \frac{a}{d} = \frac{b}{e} \)
2. \( \frac{c}{b} = \frac{f}{e} \)

These can be combined into a single condition:
\[ \frac{a}{d} = \frac{b}{e} = \frac{c}{f} \]

### Final Condition
Thus, the equations \( ax + by + c = 0 \) and \( dx + ey + f = 0 \) represent the same straight line if and only if:
\[ \frac{a}{d} = \frac{b}{e} = \frac{c}{f} \]

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