The combined equation of two lines passing through origin and each making an angle `45^(@) and 135^(@)` with the positive X axis is….

Options

A. `x^(2) + y^(2) = 0`
B. xy = 1
C. `x^(2) - y^(2) = 0`
D. `x^(2) + xy = 0`

Answer

To find the combined equation of two lines passing through the origin, each making angles of \(45^\circ\) and \(135^\circ\) with the positive X-axis, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the angles and slopes**:
- The first line makes an angle of \(45^\circ\) with the positive X-axis.
- The second line makes an angle of \(135^\circ\) with the positive X-axis.
- The slope \(M\) of a line making an angle \(\theta\) with the positive X-axis is given by \(M = \tan(\theta)\).

2. **Calculate the slopes**:
- For the first line:
\[ M_1 = \tan(45^\circ) = 1 \]
- For the second line:
\[ M_2 = \tan(135^\circ) = -1 \]

3. **Write the equations of the lines**:
- The equation of the first line (through the origin) is:
\[ y = M_1 x \implies y = 1x \implies y = x \]
- The equation of the second line (through the origin) is:
\[ y = M_2 x \implies y = -1x \implies y = -x \]

4. **Convert the equations to standard form**:
- The first line can be written as:
\[ x - y = 0 \]
- The second line can be written as:
\[ x + y = 0 \]

5. **Find the combined equation**:
- The combined equation of the two lines can be found by multiplying their individual equations:
\[ (x - y)(x + y) = 0 \]
- Using the difference of squares formula, this simplifies to:
\[ x^2 - y^2 = 0 \]

### Final Answer:
The combined equation of the two lines is:
\[ x^2 - y^2 = 0 \]