If is the line whose equation is `ax +by=c.` Let M be the reflection of 'L through the y-axis and let N be the reflection of L through the x-axis. Which of the following must be true about M and N for choices of `a, b` and `c?`

Options

A. The x- intercepts of M and N are equal
B. The y- intercepts of M and N are equal
C. The slopes of M and N are equal
D. The slopes of M and N are reciprocal

Answer

To solve the problem, we need to find the equations of the lines M and N, which are the reflections of the line L given by the equation \( ax + by = c \) through the y-axis and x-axis, respectively.

### Step 1: Write the equation of line L
The equation of line L is given as:
\[ ax + by = c \]

### Step 2: Find the reflection of line L through the y-axis (line M)
When a line is reflected through the y-axis, the x-coefficient changes sign. Therefore, the equation of line M will be:
\[ -a x + by = c \]
This can be rewritten as:
\[ -a x + by - c = 0 \]

### Step 3: Find the reflection of line L through the x-axis (line N)
When a line is reflected through the x-axis, the y-coefficient changes sign. Therefore, the equation of line N will be:
\[ ax - by = c \]
This can be rewritten as:
\[ ax - by - c = 0 \]

### Step 4: Determine the slopes of lines M and N
To find the slopes of lines M and N, we can rearrange their equations into slope-intercept form \( y = mx + b \).

For line M:
\[ by = ax + c \implies y = \frac{a}{b}x + \frac{c}{b} \]
So, the slope of line M is:
\[ m_M = \frac{a}{b} \]

For line N:
\[ -by = -ax + c \implies y = \frac{a}{b}x - \frac{c}{b} \]
So, the slope of line N is:
\[ m_N = \frac{a}{b} \]

### Step 5: Conclusion about the slopes of M and N
Since both lines M and N have the same slope:
\[ m_M = m_N = \frac{a}{b} \]
This means that the slopes of lines M and N are equal.

### Final Answer
The statement that must be true about lines M and N is that they have the same slope.

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