Consider the lines `L_(1) -=3x-4y+2=0 " and " L_(2)-=3y-4x-5=0.` Now, choose the correct statement(s).
(a)The line x+y=0 bisects the acute angle between `L_(1) "and " L_(2)` containing the origin.
(b)The line x-y+1=0 bisects the obtuse angle between `L_(1) " and " L_(2)` not containing the origin.
(c)The line x+y+3=0 bisects the obtuse angle between `L_(1) " and " L_(2)` containing the origin.
(d)The line x-y+1=0 bisects the acute angle between `L_(1) " and " L_(2)` not containing the origin.

Options

A. The line x+y=0 bisects the acute angle between `L_(1) "and " L_(2)` containing the origin.
B. The line x-y+1=0 bisects the obtuse angle between `L_(1) " and " L_(2)` not containing the origin.
C. The line x+y+3=0 bisects the obtuse angle between `L_(1) " and " L_(2)` containing the origin.
D. The line x-y+1=0 bisects the acute angle between `L_(1) " and " L_(2)` not containing the origin.

Explanation

To solve the problem, we need to analyze the given lines and determine the angle bisectors. Let's go through the solution step by step.

### Step 1: Rewrite the equations of the lines
The lines are given as:
- \( L_1: 3x - 4y + 2 = 0 \)
- \( L_2: 3y - 4x - 5 = 0 \)

We can rewrite \( L_2 \) in standard form:
\[ 4x - 3y + 5 = 0 \]

### Step 2: Identify coefficients
From the equations, we identify the coefficients:
- For \( L_1 \): \( a_1 = 3, b_1 = -4, c_1 = 2 \)
- For \( L_2 \): \( a_2 = 4, b_2 = -3, c_2 = 5 \)

### Step 3: Calculate \( a_1 a_2 + b_1 b_2 \)
Now we calculate \( a_1 a_2 + b_1 b_2 \):
\[ a_1 a_2 + b_1 b_2 = (3)(4) + (-4)(-3) = 12 + 12 = 24 \]
Since \( 24 > 0 \), this indicates that the angle bisectors will be determined by the signs of the constants \( c_1 \) and \( c_2 \).

### Step 4: Analyze the signs of \( c_1 \) and \( c_2 \)
The constants are:
- \( c_1 = 2 \) (positive)
- \( c_2 = 5 \) (positive)

Since both constants are positive, we can conclude that:
- The acute angle bisector will contain the origin.
- The obtuse angle bisector will not contain the origin.

### Step 5: Find the equations of the angle bisectors
The equations of the angle bisectors can be found using the formula:
\[ \frac{a_1 x + b_1 y + c_1}{\sqrt{a_1^2 + b_1^2}} = \pm \frac{a_2 x + b_2 y + c_2}{\sqrt{a_2^2 + b_2^2}} \]

Calculating the denominators:
\[ \sqrt{a_1^2 + b_1^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = 5 \]
\[ \sqrt{a_2^2 + b_2^2} = \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = 5 \]

### Step 6: Set up the equations
Thus, we have:
\[ \frac{3x - 4y + 2}{5} = \pm \frac{4x - 3y + 5}{5} \]

### Step 7: Solve for the acute angle bisector (positive sign)
Taking the positive sign:
\[ 3x - 4y + 2 = 4x - 3y + 5 \]
Rearranging gives:
\[ 3x - 4y + 2 - 4x + 3y - 5 = 0 \implies -x + y - 3 = 0 \implies x - y + 3 = 0 \]

### Step 8: Solve for the obtuse angle bisector (negative sign)
Taking the negative sign:
\[ 3x - 4y + 2 = - (4x - 3y + 5) \]
Rearranging gives:
\[ 3x - 4y + 2 + 4x - 3y - 5 = 0 \implies 7x - 7y - 3 = 0 \implies x - y + 1 = 0 \]

### Conclusion
The equations of the angle bisectors are:
1. For the acute angle bisector: \( x - y + 3 = 0 \) (contains the origin)
2. For the obtuse angle bisector: \( x - y + 1 = 0 \) (does not contain the origin)

### Final Statements
- Statement (a) is incorrect.
- Statement (b) is correct.
- Statement (c) is incorrect.
- Statement (d) is correct.