Find the equation of the bisector planes of the angles between the planes `2x-y+2z-19=0 and 4x-3y+12z+3=0` and specify the plane which bisects the acute angle and the planes which bisects the obtuse angle.

Answer

To find the equation of the bisector planes of the angles between the planes \(2x - y + 2z - 19 = 0\) and \(4x - 3y + 12z + 3 = 0\), we will follow these steps:

### Step 1: Identify the coefficients of the planes
The equations of the planes are:
- Plane 1: \(2x - y + 2z - 19 = 0\)
- Plane 2: \(4x - 3y + 12z + 3 = 0\)

From these equations, we can identify the coefficients:
- For Plane 1: \(a_1 = 2\), \(b_1 = -1\), \(c_1 = 2\), \(d_1 = -19\)
- For Plane 2: \(a_2 = 4\), \(b_2 = -3\), \(c_2 = 12\), \(d_2 = 3\)

### Step 2: Use the formula for the angle bisector planes
The equation of the bisector planes is given by:
\[ \frac{a_1 x + b_1 y + c_1 z + d_1}{\sqrt{a_1^2 + b_1^2 + c_1^2}} = \pm \frac{a_2 x + b_2 y + c_2 z + d_2}{\sqrt{a_2^2 + b_2^2 + c_2^2}} \]

### Step 3: Calculate the denominators
First, we calculate the denominators:
- For Plane 1:
\[ \sqrt{a_1^2 + b_1^2 + c_1^2} = \sqrt{2^2 + (-1)^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]
- For Plane 2:
\[ \sqrt{a_2^2 + b_2^2 + c_2^2} = \sqrt{4^2 + (-3)^2 + 12^2} = \sqrt{16 + 9 + 144} = \sqrt{169} = 13 \]

### Step 4: Substitute the values into the equation
Substituting the values into the bisector equation gives:
\[ \frac{2x - y + 2z - 19}{3} = \pm \frac{4x - 3y + 12z + 3}{13} \]

### Step 5: Clear the fractions
Multiply both sides by \(39\) (the least common multiple of \(3\) and \(13\)):
\[ 13(2x - y + 2z - 19) = \pm 3(4x - 3y + 12z + 3) \]

### Step 6: Expand and simplify
1. For the positive sign (acute angle):
\[ 26x - 13y + 26z - 247 = 12x - 9y + 36z + 9 \]
Rearranging gives:
\[ 26x - 12x - 13y + 9y + 26z - 36z - 247 - 9 = 0 \]
Simplifying:
\[ 14x - 4y - 10z - 256 = 0 \]

2. For the negative sign (obtuse angle):
\[ 26x - 13y + 26z - 247 = -12x + 9y - 36z - 9 \]
Rearranging gives:
\[ 26x + 12x - 13y - 9y + 26z + 36z - 247 + 9 = 0 \]
Simplifying:
\[ 38x - 22y + 62z - 238 = 0 \]

### Final Results
The equations of the bisector planes are:
- For the acute angle: \(14x - 4y - 10z - 256 = 0\)
- For the obtuse angle: \(38x - 22y + 62z - 238 = 0\)