Find the equations of the bisectors of the angles, between the lines through `(0,0)` with slopes `1` and `2`.

Answer

To find the equations of the bisectors of the angles between the lines through the origin (0,0) with slopes 1 and 2, we can follow these steps:

### Step 1: Write the equations of the lines
The equations of the lines can be expressed in slope-intercept form (y = mx).

For the line with slope 1:
- The equation is \( y = 1x \) or simply \( y = x \).

For the line with slope 2:
- The equation is \( y = 2x \).

### Step 2: Convert the equations to standard form
We can convert these equations into the standard form \( Ax + By + C = 0 \).

For the line \( y = x \):
- Rearranging gives us \( x - y = 0 \).
- Here, \( A_1 = 1, B_1 = -1, C_1 = 0 \).

For the line \( y = 2x \):
- Rearranging gives us \( 2x - y = 0 \).
- Here, \( A_2 = 2, B_2 = -1, C_2 = 0 \).

### Step 3: Use the angle bisector formula
The angle bisectors of two lines given in the form \( A_1x + B_1y + C_1 = 0 \) and \( A_2x + B_2y + C_2 = 0 \) can be found using the formula:

\[ \frac{A_1x + B_1y + C_1}{\sqrt{A_1^2 + B_1^2}} = \pm \frac{A_2x + B_2y + C_2}{\sqrt{A_2^2 + B_2^2}} \]

### Step 4: Substitute the values into the formula
Substituting the values we found:

\[ \frac{1x - 1y + 0}{\sqrt{1^2 + (-1)^2}} = \pm \frac{2x - 1y + 0}{\sqrt{2^2 + (-1)^2}} \]

Calculating the denominators:
- For the first line: \( \sqrt{1 + 1} = \sqrt{2} \)
- For the second line: \( \sqrt{4 + 1} = \sqrt{5} \)

Thus, we have:

\[ \frac{x - y}{\sqrt{2}} = \pm \frac{2x - y}{\sqrt{5}} \]

### Step 5: Separate into two equations
This gives us two equations:

1. \( \frac{x - y}{\sqrt{2}} = \frac{2x - y}{\sqrt{5}} \)
2. \( \frac{x - y}{\sqrt{2}} = -\frac{2x - y}{\sqrt{5}} \)

### Step 6: Solve the first equation
Cross-multiplying the first equation:

\[ (x - y) \sqrt{5} = (2x - y) \sqrt{2} \]

Expanding gives:

\[ \sqrt{5}x - \sqrt{5}y = 2\sqrt{2}x - \sqrt{2}y \]

Rearranging terms:

\[ \sqrt{5}x - 2\sqrt{2}x + \sqrt{5}y - \sqrt{2}y = 0 \]

This simplifies to:

\[ (\sqrt{5} - 2\sqrt{2})x + (\sqrt{5} - \sqrt{2})y = 0 \]

### Step 7: Solve the second equation
Cross-multiplying the second equation:

\[ (x - y) \sqrt{5} = -(2x - y) \sqrt{2} \]

Expanding gives:

\[ \sqrt{5}x - \sqrt{5}y = -2\sqrt{2}x + \sqrt{2}y \]

Rearranging terms:

\[ \sqrt{5}x + 2\sqrt{2}x - \sqrt{5}y - \sqrt{2}y = 0 \]

This simplifies to:

\[ (\sqrt{5} + 2\sqrt{2})x + (-\sqrt{5} - \sqrt{2})y = 0 \]

### Final Equations of the Bisectors
Thus, the equations of the angle bisectors are:

1. \((\sqrt{5} - 2\sqrt{2})x + (\sqrt{5} - \sqrt{2})y = 0\)
2. \((\sqrt{5} + 2\sqrt{2})x + (-\sqrt{5} - \sqrt{2})y = 0\)