A ray of light is sent along the line which passes through the point (2, 3). The ray is reflected from the point P on x-axis. If the reflected ray passes through the point (6, 4), then the co-ordinates of P are

Options

A. `((26)/(7),0)`
B. `(0,(26)/(7))`
C. `((-26)/(7),0)`
D. `(-3,0)`

Answer

To solve the problem, we need to find the coordinates of point P on the x-axis where a ray of light reflects. The ray of light is initially sent from the point (2, 3) and reflects off point P before passing through the point (6, 4).

### Step-by-Step Solution:

1. **Identify Coordinates of Point P**:
Since point P lies on the x-axis, its coordinates can be represented as \( P(a, 0) \), where \( a \) is the x-coordinate we need to find.

2. **Find the Slope of the Incident Ray**:
The incident ray travels from (2, 3) to \( P(a, 0) \). The slope \( m_1 \) of the line can be calculated using the formula:
\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - 3}{a - 2} = \frac{-3}{a - 2} \]

3. **Find the Slope of the Reflected Ray**:
The reflected ray travels from \( P(a, 0) \) to (6, 4). The slope \( m_2 \) of this line is:
\[ m_2 = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 0}{6 - a} = \frac{4}{6 - a} \]

4. **Apply the Law of Reflection**:
According to the law of reflection, the angle of incidence is equal to the angle of reflection. This means:
\[ m_1 = -m_2 \]
Therefore, we can write:
\[ \frac{-3}{a - 2} = -\frac{4}{6 - a} \]

5. **Cross Multiply to Solve for a**:
Cross multiplying gives us:
\[ -3(6 - a) = -4(a - 2) \]
Simplifying this equation:
\[ -18 + 3a = -4a + 8 \]
Rearranging terms:
\[ 3a + 4a = 18 + 8 \]
\[ 7a = 26 \]
\[ a = \frac{26}{7} \]

6. **Determine the Coordinates of Point P**:
Since we found \( a \), the coordinates of point P are:
\[ P\left(\frac{26}{7}, 0\right) \]

### Final Answer:
The coordinates of point P are \( \left(\frac{26}{7}, 0\right) \).