Find equations of acute and obtuse angle bisectors of the angle between the lines `4x+ 3y-7 =0 and 24x + 7y-31 = 0`. Also comment in which bisector origin lies.

Options

A. bisector of the obtuse angle between them is `x-2y+1=0`
B. bisector of the obtuse angle between them is `2x-y+3=0`
C. bisector of the angle containing origin is `x-2y+1=0`
D. bisector of the angle containing the point is `(1,-2)` is `x-2y+1=0`

Answer

To find the equations of the acute and obtuse angle bisectors of the angle between the lines \(4x + 3y - 7 = 0\) and \(24x + 7y - 31 = 0\), we can use the formula for the angle bisectors of two lines given by:

\[ \frac{Ax + By + C_1}{\sqrt{A^2 + B^2}} = \pm \frac{Px + Qy + C_2}{\sqrt{P^2 + Q^2}} \]

where \(A, B, C_1\) are the coefficients of the first line and \(P, Q, C_2\) are the coefficients of the second line.

### Step 1: Identify coefficients
From the first line \(4x + 3y - 7 = 0\):
- \(A = 4\)
- \(B = 3\)
- \(C_1 = -7\)

From the second line \(24x + 7y - 31 = 0\):
- \(P = 24\)
- \(Q = 7\)
- \(C_2 = -31\)

### Step 2: Calculate the denominators
Calculate \(\sqrt{A^2 + B^2}\) and \(\sqrt{P^2 + Q^2}\):
\[ \sqrt{A^2 + B^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]
\[ \sqrt{P^2 + Q^2} = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \]

### Step 3: Set up the angle bisector equations
Using the angle bisector formula:
\[ \frac{4x + 3y - 7}{5} = \pm \frac{24x + 7y - 31}{25} \]

### Step 4: Multiply through by the denominators
Multiply both sides by \(125\) (the least common multiple of 5 and 25):
\[ 25(4x + 3y - 7) = \pm 5(24x + 7y - 31) \]

### Step 5: Expand both sides
For the positive case:
\[ 100x + 75y - 175 = 120x + 35y - 155 \]
Rearranging gives:
\[ 100x - 120x + 75y - 35y = 175 - 155 \]
\[ -20x + 40y = 20 \]
Dividing by -20:
\[ x - 2y + 1 = 0 \quad \text{(Acute angle bisector)} \]

For the negative case:
\[ 100x + 75y - 175 = -5(24x + 7y - 31) \]
Expanding gives:
\[ 100x + 75y - 175 = -120x - 35y + 155 \]
Rearranging gives:
\[ 100x + 120x + 75y + 35y = 175 + 155 \]
\[ 220x + 110y = 330 \]
Dividing by 110:
\[ 2x + y - 3 = 0 \quad \text{(Obtuse angle bisector)} \]

### Step 6: Comment on the position of the origin
To determine which bisector the origin lies on, we can substitute \(x = 0\) and \(y = 0\) into both bisector equations.

1. For the acute angle bisector \(x - 2y + 1 = 0\):
\[ 0 - 2(0) + 1 = 1 \quad \text{(positive)} \]

2. For the obtuse angle bisector \(2x + y - 3 = 0\):
\[ 2(0) + 0 - 3 = -3 \quad \text{(negative)} \]

Since the acute angle bisector evaluates to a positive value and the obtuse angle bisector evaluates to a negative value, the origin lies on the obtuse angle bisector.

### Final Equations
- Acute angle bisector: \(x - 2y + 1 = 0\)
- Obtuse angle bisector: \(2x + y - 3 = 0\)