Find the equations to the straight lines bisecting the angles between the following pairs of straight lines, placing first the bisector of the angle in which the origin lies.
`2x+ y= 4 and y+ 3x= 5`.

Answer

To find the equations of the straight lines bisecting the angles between the given pairs of straight lines \(2x + y = 4\) and \(y + 3x = 5\), we will follow these steps:

### Step 1: Write the equations in standard form
The given equations can be rewritten in the standard form \(Ax + By + C = 0\):
1. For \(2x + y = 4\):
\[ 2x + y - 4 = 0 \quad \text{(Let this be } L_1\text{)} \]
2. For \(y + 3x = 5\):
\[ 3x + y - 5 = 0 \quad \text{(Let this be } L_2\text{)} \]

### Step 2: Identify coefficients
From the equations, we can identify:
- For \(L_1\): \(A_1 = 2\), \(B_1 = 1\), \(C_1 = -4\)
- For \(L_2\): \(A_2 = 3\), \(B_2 = 1\), \(C_2 = -5\)

### Step 3: Check the position of the origin
To determine which angle bisector is closer to the origin, we evaluate \(L_1\) and \(L_2\) at the origin \((0, 0)\):
- For \(L_1\):
\[ L_1(0, 0) = 2(0) + 1(0) - 4 = -4 \]
- For \(L_2\):
\[ L_2(0, 0) = 3(0) + 1(0) - 5 = -5 \]

### Step 4: Calculate the product
Now, we calculate the product of the values:
\[ L_1(0, 0) \cdot L_2(0, 0) = (-4) \cdot (-5) = 20 \]
Since the product is positive, the first angle bisector will be the positive one.

### Step 5: Use the angle bisector formula
The angle bisector formula is given by:
\[ \frac{A_1 x + B_1 y + C_1}{\sqrt{A_1^2 + B_1^2}} = \pm \frac{A_2 x + B_2 y + C_2}{\sqrt{A_2^2 + B_2^2}} \]
Substituting the values:
\[ \frac{2x + y - 4}{\sqrt{2^2 + 1^2}} = \frac{3x + y - 5}{\sqrt{3^2 + 1^2}} \]

### Step 6: Simplify the equation
Calculating the denominators:
\[ \sqrt{2^2 + 1^2} = \sqrt{5}, \quad \sqrt{3^2 + 1^2} = \sqrt{10} \]
Thus, we have:
\[ \frac{2x + y - 4}{\sqrt{5}} = \frac{3x + y - 5}{\sqrt{10}} \]

### Step 7: Cross-multiply and simplify
Cross-multiplying gives:
\[ \sqrt{10}(2x + y - 4) = \sqrt{5}(3x + y - 5) \]
Expanding both sides:
\[ 2\sqrt{10}x + \sqrt{10}y - 4\sqrt{10} = 3\sqrt{5}x + \sqrt{5}y - 5\sqrt{5} \]

### Step 8: Rearranging the equation
Rearranging terms leads to:
\[ (2\sqrt{10} - 3\sqrt{5})x + (\sqrt{10} - \sqrt{5})y + (5\sqrt{5} - 4\sqrt{10}) = 0 \]

### Step 9: Find the second angle bisector
For the second angle bisector, we use the negative sign in the angle bisector formula:
\[ \frac{2x + y - 4}{\sqrt{5}} = -\frac{3x + y - 5}{\sqrt{10}} \]
Following similar steps as above will yield the second bisector's equation.

### Final Result
The equations of the angle bisectors are:
1. First bisector (towards the origin):
\[ (2\sqrt{10} - 3\sqrt{5})x + (\sqrt{10} - \sqrt{5})y + (5\sqrt{5} - 4\sqrt{10}) = 0 \]
2. Second bisector:
\[ (2\sqrt{10} + 3\sqrt{5})x + (\sqrt{10} + \sqrt{5})y + (5\sqrt{5} + 4\sqrt{10}) = 0 \]