Find the bisector of the obtuse angle between the lines `12x+5y-4=0` and `3x+4y+7=0`

Answer

To find the bisector of the obtuse angle between the lines \(12x + 5y - 4 = 0\) and \(3x + 4y + 7 = 0\), we can follow these steps:

### Step 1: Identify the coefficients
The given lines can be expressed in the form \(Ax + By + C = 0\):
- For the first line \(12x + 5y - 4 = 0\), we have:
- \(a_1 = 12\), \(b_1 = 5\), \(c_1 = -4\)
- For the second line \(3x + 4y + 7 = 0\), we have:
- \(a_2 = 3\), \(b_2 = 4\), \(c_2 = 7\)

### Step 2: Check the condition for obtuse angle
To find the bisector of the obtuse angle, we need to check the condition:
\[ a_1 a_2 + b_1 b_2 \leq 0 \]
Calculating:
\[ a_1 a_2 = 12 \times 3 = 36 \]
\[ b_1 b_2 = 5 \times 4 = 20 \]
Thus,
\[ a_1 a_2 + b_1 b_2 = 36 + 20 = 56 > 0 \]
Since this is greater than 0, we need to modify one of the equations. We can multiply the first line by -1:
\[ -12x - 5y + 4 = 0 \]

### Step 3: Recalculate the coefficients
Now we have:
- For the modified first line:
- \(a_1 = -12\), \(b_1 = -5\), \(c_1 = 4\)
- For the second line:
- \(a_2 = 3\), \(b_2 = 4\), \(c_2 = 7\)

### Step 4: Check the condition again
Now, we recalculate:
\[ a_1 a_2 + b_1 b_2 = (-12)(3) + (-5)(4) = -36 - 20 = -56 < 0 \]
This confirms that we can now find the obtuse angle bisector.

### Step 5: Set up the angle bisector equation
The equation for the angle bisector is given by:
\[ \frac{a_1 x + b_1 y + c_1}{\sqrt{a_1^2 + b_1^2}} = \pm \frac{a_2 x + b_2 y + c_2}{\sqrt{a_2^2 + b_2^2}} \]

### Step 6: Calculate the left side
Calculating \( \sqrt{a_1^2 + b_1^2} \):
\[ \sqrt{(-12)^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} = 13 \]
So the left side becomes:
\[ \frac{-12x - 5y + 4}{13} \]

### Step 7: Calculate the right side
Calculating \( \sqrt{a_2^2 + b_2^2} \):
\[ \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
So the right side becomes:
\[ \frac{3x + 4y + 7}{5} \]

### Step 8: Set up the equation
Now we have:
\[ \frac{-12x - 5y + 4}{13} = -\frac{3x + 4y + 7}{5} \]

### Step 9: Cross-multiply
Cross-multiplying gives:
\[ 5(-12x - 5y + 4) = -13(3x + 4y + 7) \]
Expanding both sides:
\[ -60x - 25y + 20 = -39x - 52y - 91 \]

### Step 10: Rearranging the equation
Rearranging gives:
\[ -60x + 39x - 25y + 52y + 20 + 91 = 0 \]
This simplifies to:
\[ -21x + 27y + 111 = 0 \]
Or, multiplying through by -1:
\[ 21x - 27y - 111 = 0 \]

### Final Equation
Thus, the equation of the bisector of the obtuse angle between the two lines is:
\[ 21x - 27y + 111 = 0 \]