Three straight lines `2x+11y-5=0, 24x+7y-20=0 and 4x-3y-2=0`

Options

A. form a triangle
B. are only concurrent
C. are concurrent with one line bisecting the angle between the other two
D. None of the above

Answer

To determine the relationship between the three lines given by the equations \(2x + 11y - 5 = 0\), \(24x + 7y - 20 = 0\), and \(4x - 3y - 2 = 0\), we will check if they are concurrent and if one of them bisects the angle between the other two.

### Step 1: Identify the equations of the lines
The equations of the three lines are:
1. Line 1: \(L_1: 2x + 11y - 5 = 0\)
2. Line 2: \(L_2: 24x + 7y - 20 = 0\)
3. Line 3: \(L_3: 4x - 3y - 2 = 0\)

**Hint:** Write down the equations clearly to avoid confusion later.

### Step 2: Use the angle bisector condition
To check if one line bisects the angle between the other two, we can use the angle bisector theorem. The condition 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 \(a_1, b_1, c_1\) are the coefficients from one line, and \(a_2, b_2, c_2\) are from the other line.

### Step 3: Apply the formula
Let's apply this to \(L_2\) and \(L_3\):

1. For \(L_2: 24x + 7y - 20 = 0\), we have:
- \(a_1 = 24\), \(b_1 = 7\), \(c_1 = -20\)

2. For \(L_3: 4x - 3y - 2 = 0\), we have:
- \(a_2 = 4\), \(b_2 = -3\), \(c_2 = -2\)

Now we compute:

\[ \frac{24x + 7y - 20}{\sqrt{24^2 + 7^2}} = \pm \frac{4x - 3y - 2}{\sqrt{4^2 + (-3)^2}} \]

### Step 4: Calculate the denominators
Calculate the denominators:

1. For \(L_2\):
\[ \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \]

2. For \(L_3\):
\[ \sqrt{4^2 + (-3)^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

### Step 5: Set up the equation
Now substituting back into the equation:

\[ \frac{24x + 7y - 20}{25} = \pm \frac{4x - 3y - 2}{5} \]

### Step 6: Solve for the positive case
Taking the positive case:

\[ 24x + 7y - 20 = 5(4x - 3y - 2) \]

Expanding the right side:

\[ 24x + 7y - 20 = 20x - 15y - 10 \]

### Step 7: Rearranging the equation
Rearranging gives:

\[ 24x - 20x + 7y + 15y - 20 + 10 = 0 \]

This simplifies to:

\[ 4x + 22y - 10 = 0 \]

### Step 8: Compare with the first line
Now, we can divide the entire equation by 2:

\[ 2x + 11y - 5 = 0 \]

This is exactly the same as our first line \(L_1\).

### Conclusion
Since we have shown that the line \(L_1\) is the angle bisector of lines \(L_2\) and \(L_3\), we conclude that the three lines are concurrent with one line bisecting the angle between the other two.

**Final Answer:** The lines are concurrent with one line bisecting the angle between the other two.