The equation of the bisectors of the angles between the two intersecting lines `(x-3)/(cos theta ) = (y+5)/(sin theta) and (x-3)/(cos theta) = (y+5)/(sin theta)` are` (x-3)/(cos alpha) = (y+5)/(sin alpha) and (x-3)/beta = (y+5)/gamma ,` then

Options

A. `alpha=(theta+phi)/2`
B. `beta =-sin alpha`
C. `gamma = cos alpha`
D. `beta = sin alpha`

Answer

To solve the problem, we need to find the equations of the angle bisectors of the two given lines and compare them with the provided equations. Let's break it down step by step.

### Step 1: Understand the given equations
The equations of the two lines are:
1. \(\frac{x-3}{\cos \theta} = \frac{y+5}{\sin \theta}\)
2. \(\frac{x-3}{\cos \theta} + \frac{y+5}{\sin \theta} = 0\)

### Step 2: Rewrite the equations in slope-intercept form
From the first equation, we can express \(y\) in terms of \(x\):
\[ y + 5 = \frac{\sin \theta}{\cos \theta}(x - 3) \]
\[ y = \tan \theta (x - 3) - 5 \]
This represents the first line.

For the second equation:
\[ \frac{x-3}{\cos \theta} + \frac{y+5}{\sin \theta} = 0 \]
Rearranging gives:
\[ \frac{y+5}{\sin \theta} = -\frac{x-3}{\cos \theta} \]
\[ y + 5 = -\tan \theta (x - 3) \]
\[ y = -\tan \theta (x - 3) - 5 \]
This represents the second line.

### Step 3: Find the angle bisectors
The angle bisectors of the two lines can be found using the formula:
\[ \frac{x - x_1}{\cos \alpha} = \frac{y - y_1}{\sin \alpha} \]
where \(\alpha\) is the angle made by the bisector with the x-axis.

The first bisector makes an angle of \(\frac{\theta + \phi}{2}\) with the x-axis, so:
\[ \frac{x - 3}{\cos(\frac{\theta + \phi}{2})} = \frac{y + 5}{\sin(\frac{\theta + \phi}{2})} \]

The second bisector makes an angle of \(90 + \frac{\theta + \phi}{2}\) with the x-axis, so:
\[ \frac{x - 3}{\cos(90 + \frac{\theta + \phi}{2})} = \frac{y + 5}{\sin(90 + \frac{\theta + \phi}{2})} \]
Using the identities \(\cos(90 + x) = -\sin x\) and \(\sin(90 + x) = \cos x\):
\[ \frac{x - 3}{-\sin(\frac{\theta + \phi}{2})} = \frac{y + 5}{\cos(\frac{\theta + \phi}{2})} \]

### Step 4: Compare with given equations
We are given that the bisectors can be represented as:
1. \(\frac{x-3}{\cos \alpha} = \frac{y+5}{\sin \alpha}\)
2. \(\frac{x-3}{\beta} = \frac{y+5}{\gamma}\)

From our derivation:
- For the first bisector, we have \(\alpha = \frac{\theta + \phi}{2}\).
- For the second bisector, we have \(\beta = -\sin(\frac{\theta + \phi}{2})\) and \(\gamma = \cos(\frac{\theta + \phi}{2})\).

### Step 5: Final relationships
Thus, we can conclude:
- \(\alpha = \frac{\theta + \phi}{2}\)
- \(\beta = -\sin(\alpha)\)
- \(\gamma = \cos(\alpha)\)

### Summary
The relationships we found are:
- \(\alpha = \theta + \frac{\phi}{2}\)
- \(\beta = -\sin(\alpha)\)
- \(\gamma = \cos(\alpha)\)