Joint equation of lines bisecting angles between co-oridnates axes is

Options

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

Answer

To find the joint equation of the lines bisecting the angles between the coordinate axes, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Problem**:
We need to find the joint equation of the lines that bisect the angles formed by the coordinate axes (x-axis and y-axis). There are two lines that bisect these angles.

2. **Identifying the Angles**:
The coordinate axes form four angles:
- The first angle (between the positive x-axis and positive y-axis) is 90 degrees.
- The bisector of this angle will be at 45 degrees.
- The second angle (between the positive y-axis and negative x-axis) is also 90 degrees, and its bisector will be at 135 degrees.

3. **Finding the Slopes of the Bisectors**:
- The slope of the line that bisects the angle between the positive x-axis and positive y-axis (45 degrees) is given by:
\[ m_1 = \tan(45^\circ) = 1 \]
- The slope of the line that bisects the angle between the positive y-axis and negative x-axis (135 degrees) is given by:
\[ m_2 = \tan(135^\circ) = \tan(180^\circ - 45^\circ) = -\tan(45^\circ) = -1 \]

4. **Writing the Equations of the Lines**:
- The equation of the first line (slope = 1) passing through the origin (0,0) is:
\[ y = x \quad \text{or} \quad x - y = 0 \]
- The equation of the second line (slope = -1) passing through the origin is:
\[ y = -x \quad \text{or} \quad x + y = 0 \]

5. **Finding the Joint Equation**:
- The joint equation of the two lines can be found by multiplying their equations:
\[ (x - y)(x + y) = 0 \]
- Expanding this gives:
\[ x^2 - y^2 = 0 \]

6. **Final Result**:
The joint equation of the lines bisecting the angles between the coordinate axes is:
\[ x^2 - y^2 = 0 \]