The incident ray is along the line `3x-4y-3=0` and the reflected ray is along the line `24 x+7y+5=0.` Find the equation of mirrors.

Explanation

To find the equation of the mirrors given the incident ray and reflected ray, we will use the concept of angle bisectors. Here are the steps to solve the problem:

### Step 1: Identify the equations of the incident and reflected rays
The incident ray is given by the equation:
\[ 3x - 4y - 3 = 0 \]

The reflected ray is given by the equation:
\[ 24x + 7y + 5 = 0 \]

### Step 2: Write the equations in the standard form
The equations are already in the standard form \( Ax + By + C = 0 \):
- For the incident ray: \( A_1 = 3, B_1 = -4, C_1 = -3 \)
- For the reflected ray: \( A_2 = 24, B_2 = 7, C_2 = 5 \)

### Step 3: Use the angle bisector formula
The angle bisector of two lines can be found using the formula:
\[ \frac{A_1x + B_1y + C_1}{\sqrt{A_1^2 + B_1^2}} = \pm \frac{A_2x + B_2y + C_2}{\sqrt{A_2^2 + B_2^2}} \]

### Step 4: Calculate the necessary components
First, calculate the denominators:
- For the incident ray:
\[ \sqrt{A_1^2 + B_1^2} = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

- For the reflected ray:
\[ \sqrt{A_2^2 + B_2^2} = \sqrt{24^2 + 7^2} = \sqrt{576 + 49} = \sqrt{625} = 25 \]

### Step 5: Set up the angle bisector equations
Substituting into the angle bisector formula gives us:
\[ \frac{3x - 4y - 3}{5} = \pm \frac{24x + 7y + 5}{25} \]

### Step 6: Solve for the positive case
Taking the positive case:
\[ \frac{3x - 4y - 3}{5} = \frac{24x + 7y + 5}{25} \]

Cross-multiplying gives:
\[ 25(3x - 4y - 3) = 5(24x + 7y + 5) \]
Expanding both sides:
\[ 75x - 100y - 75 = 120x + 35y + 25 \]

Rearranging terms:
\[ 75x - 120x - 100y - 35y - 75 - 25 = 0 \]
This simplifies to:
\[ -45x - 135y - 100 = 0 \]
Dividing by -5:
\[ 9x + 27y + 20 = 0 \]

### Step 7: Solve for the negative case
Now, taking the negative case:
\[ \frac{3x - 4y - 3}{5} = -\frac{24x + 7y + 5}{25} \]

Cross-multiplying gives:
\[ 25(3x - 4y - 3) = -5(24x + 7y + 5) \]
Expanding both sides:
\[ 75x - 100y - 75 = -120x - 35y - 25 \]

Rearranging terms:
\[ 75x + 120x - 100y + 35y - 75 + 25 = 0 \]
This simplifies to:
\[ 195x - 65y - 50 = 0 \]
Dividing by 5:
\[ 39x - 13y - 10 = 0 \]

### Final Result
The equations of the mirrors (angle bisectors) are:
1. \( 9x + 27y + 20 = 0 \)
2. \( 39x - 13y - 10 = 0 \)