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.
`4x+ 3y- 7=0 and 24x+ 7y- 31= 0`

Answer

To find the equations of the straight lines bisecting the angles between the given pairs of straight lines, we will follow these steps:

### Step 1: Identify the equations of the lines
The equations of the lines are:
- \( L_1: 4x + 3y - 7 = 0 \)
- \( L_2: 24x + 7y - 31 = 0 \)

### Step 2: Evaluate the lines at the origin
We will check the values of \( L_1 \) and \( L_2 \) at the origin (0, 0):
- For \( L_1 \):
\[ L_1(0, 0) = 4(0) + 3(0) - 7 = -7 \]
- For \( L_2 \):
\[ L_2(0, 0) = 24(0) + 7(0) - 31 = -31 \]

### Step 3: Determine the product of the values
Now, we calculate the product of the values obtained:
\[ L_1(0, 0) \times L_2(0, 0) = (-7) \times (-31) = 217 \]
Since the product is positive, the origin lies between the two lines, and we can find the first angle bisector.

### Step 4: Use the angle bisector formula
The formula for the angle bisector is given by:
\[ \frac{A_1x + B_1y + C_1}{\sqrt{A_1^2 + B_1^2}} = \pm \frac{A_2x + B_2y + C_2}{\sqrt{A_2^2 + B_2^2}} \]
Where:
- For \( L_1 \): \( A_1 = 4, B_1 = 3, C_1 = -7 \)
- For \( L_2 \): \( A_2 = 24, B_2 = 7, C_2 = -31 \)

### Step 5: Calculate the left-hand and right-hand sides
First, we calculate the denominators:
\[ \sqrt{A_1^2 + B_1^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]
\[ \sqrt{A_2^2 + B_2^2} = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \]

### Step 6: Substitute into the angle bisector formula
Substituting into the angle bisector formula:
\[ \frac{4x + 3y - 7}{5} = \frac{24x + 7y - 31}{25} \]

### Step 7: Cross-multiply to eliminate the fractions
Cross-multiplying gives:
\[ 25(4x + 3y - 7) = 5(24x + 7y - 31) \]
Expanding both sides:
\[ 100x + 75y - 175 = 120x + 35y - 155 \]

### Step 8: Rearranging the equation
Rearranging the equation gives:
\[ 100x + 75y - 120x - 35y = -155 + 175 \]
\[ -20x + 40y = 20 \]
Dividing through by -20:
\[ x - 2y + 1 = 0 \]

### Step 9: Find the second angle bisector
For the second angle bisector, we use the negative sign in the angle bisector formula:
\[ \frac{4x + 3y - 7}{5} = -\frac{24x + 7y - 31}{25} \]
Cross-multiplying gives:
\[ 25(4x + 3y - 7) = -5(24x + 7y - 31) \]
Expanding both sides:
\[ 100x + 75y - 175 = -120x - 35y + 155 \]

### Step 10: Rearranging the second equation
Rearranging gives:
\[ 100x + 75y + 120x + 35y = 155 + 175 \]
\[ 220x + 110y = 330 \]
Dividing through by 110:
\[ 2x + y - 3 = 0 \]

### Final Result
The equations of the angle bisectors are:
1. \( x - 2y + 1 = 0 \) (the bisector in which the origin lies)
2. \( 2x + y - 3 = 0 \)