A ray of light through B(3,2) is reflected at the point `A(0,beta)` on the y-axis and passes through C(4,3). Then `beta` is

Options

A. `(7)/(11)`
B. `(13)/(7)`
C. `(17)/(7)`
D. `(8)/(11)`

Answer

To solve the problem step by step, we need to find the value of \( \beta \) where a ray of light reflects off the y-axis at point \( A(0, \beta) \) after passing through point \( B(3, 2) \) and before passing through point \( C(4, 3) \).

### Step 1: Understand the Reflection
The ray of light passes through point \( B(3, 2) \) and reflects at point \( A(0, \beta) \) on the y-axis before passing through point \( C(4, 3) \). The key concept here is that the angle of incidence is equal to the angle of reflection.

### Step 2: Find the Slope of Line \( AB \)
The slope of the line segment \( AB \) can be calculated using the coordinates of points \( A(0, \beta) \) and \( B(3, 2) \):
\[ \text{slope of } AB = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - \beta}{3 - 0} = \frac{2 - \beta}{3} \]

### Step 3: Find the Slope of Line \( AC \)
Next, we find the slope of the line segment \( AC \) using points \( A(0, \beta) \) and \( C(4, 3) \):
\[ \text{slope of } AC = \frac{3 - \beta}{4 - 0} = \frac{3 - \beta}{4} \]

### Step 4: Set Up the Reflection Condition
According to the law of reflection, the slopes of lines \( AB \) and \( AC \) must satisfy:
\[ \text{slope of } AB = -\text{slope of } AC \]
This gives us the equation:
\[ \frac{2 - \beta}{3} = -\frac{3 - \beta}{4} \]

### Step 5: Cross-Multiply to Solve for \( \beta \)
Cross-multiplying gives:
\[ 4(2 - \beta) = -3(3 - \beta) \]
Expanding both sides:
\[ 8 - 4\beta = -9 + 3\beta \]

### Step 6: Combine Like Terms
Now, we combine like terms:
\[ 8 + 9 = 4\beta + 3\beta \]
\[ 17 = 7\beta \]

### Step 7: Solve for \( \beta \)
Dividing both sides by 7 gives:
\[ \beta = \frac{17}{7} \]

### Final Answer
Thus, the value of \( \beta \) is:
\[ \beta = \frac{17}{7} \]