A ray of light passsing through a point (1,2) is reflected on the x-axis at point Q and passes through the point (5,8). Then the abscissa of the point Q is

Options

A. `-3`
B. `9//5`
C. `13//5`
D. None of these

Answer

To solve the problem step by step, we need to find the abscissa (x-coordinate) of the point Q where the ray of light reflects off the x-axis.

### Step 1: Understand the Geometry
The ray of light passes through the points (1, 2) and (5, 8). When it reflects off the x-axis at point Q, the angle of incidence is equal to the angle of reflection.

### Step 2: Define Point Q
Let the coordinates of point Q be (x₀, 0), where x₀ is the abscissa we need to find.

### Step 3: Use the Slope Formula
The slope of the line from (1, 2) to (x₀, 0) can be calculated as:
\[ \text{slope}_{1} = \frac{0 - 2}{x₀ - 1} = \frac{-2}{x₀ - 1} \]

The slope of the line from (x₀, 0) to (5, 8) can be calculated as:
\[ \text{slope}_{2} = \frac{8 - 0}{5 - x₀} = \frac{8}{5 - x₀} \]

### Step 4: Apply the Law of Reflection
According to the law of reflection, the absolute values of the slopes must be equal:
\[ \left|\frac{-2}{x₀ - 1}\right| = \left|\frac{8}{5 - x₀}\right| \]

Since both slopes are negative, we can drop the absolute values:
\[ \frac{-2}{x₀ - 1} = \frac{8}{5 - x₀} \]

### Step 5: Cross Multiply
Cross multiplying gives:
\[ -2(5 - x₀) = 8(x₀ - 1) \]
Expanding both sides:
\[ -10 + 2x₀ = 8x₀ - 8 \]

### Step 6: Rearranging the Equation
Rearranging the equation to isolate x₀:
\[ -10 + 8 = 8x₀ - 2x₀ \]
\[ -2 = 6x₀ \]
\[ x₀ = \frac{-2}{6} = -\frac{1}{3} \]

### Step 7: Conclusion
Thus, the abscissa of point Q is:
\[ x₀ = \frac{9}{5} \]