The equation of the bisector of that angle between the lines `x+2y-11=0,3x-6y-5=0` which contains the point `(1,-3)` is `(3x=19` (b) `3y=7` `3x=19a n d3y=7` (d) None of these

Answer

To find the equation of the angle bisector between the lines \(x + 2y - 11 = 0\) and \(3x - 6y - 5 = 0\) that contains the point \((1, -3)\), we will follow these steps:

### Step 1: Rewrite the equations in slope-intercept form
The first line is:
\[ x + 2y - 11 = 0 \implies 2y = -x + 11 \implies y = -\frac{1}{2}x + \frac{11}{2} \]
The second line is:
\[ 3x - 6y - 5 = 0 \implies -6y = -3x + 5 \implies y = \frac{1}{2}x - \frac{5}{6} \]

### Step 2: Identify the slopes of the lines
From the equations:
- The slope of the first line \(m_1 = -\frac{1}{2}\)
- The slope of the second line \(m_2 = \frac{1}{2}\)

### Step 3: Find the angle bisector
The angle bisector can be found using the formula:
\[ \frac{y - y_1}{y_2 - y_1} = \frac{m_1 + m_2}{1 - m_1 m_2} \]
However, since we need to find the specific bisector that contains the point \((1, -3)\), we will use the following approach.

### Step 4: Find the intersection point of the two lines
To find the intersection, we solve the equations:
1. \(x + 2y = 11\)
2. \(3x - 6y = 5\)

From the first equation, we can express \(y\):
\[ y = \frac{11 - x}{2} \]
Substituting this into the second equation:
\[ 3x - 6\left(\frac{11 - x}{2}\right) = 5 \]
Multiplying through by 2 to eliminate the fraction:
\[ 6x - 6(11 - x) = 10 \implies 6x - 66 + 6x = 10 \implies 12x = 76 \implies x = \frac{76}{12} = \frac{19}{3} \]
Now substituting \(x = \frac{19}{3}\) back into \(y = \frac{11 - x}{2}\):
\[ y = \frac{11 - \frac{19}{3}}{2} = \frac{\frac{33 - 19}{3}}{2} = \frac{\frac{14}{3}}{2} = \frac{7}{3} \]
Thus, the intersection point is \(\left(\frac{19}{3}, \frac{7}{3}\right)\).

### Step 5: Find the equation of the angle bisector
The angle bisector can be expressed as:
\[ \frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} \]
Using the coordinates of the intersection point \(\left(\frac{19}{3}, \frac{7}{3}\right)\) and the point \((1, -3)\).

### Step 6: Check which bisector contains the point (1, -3)
To find the correct bisector, we can check the equations of the bisectors derived from the slopes. The equations of the angle bisectors are:
1. \(3x = 19\)
2. \(3y = 7\)

We need to check which of these contains the point \((1, -3)\):
- For \(3x = 19\): \(3(1) = 3 \neq 19\)
- For \(3y = 7\): \(3(-3) = -9 \neq 7\)

Thus, the equation of the angle bisector that contains the point \((1, -3)\) is \(3x = 19\).

### Final Answer
The equation of the bisector is:
\[ \text{(a) } 3x = 19 \]