A ray of light coming from the point (1,2) is reflected at a pont A on the x-axis and then passes through the point (5,3). The co ordinate of the point A is

Options

A. `(13/5,0)`
B. `(5/13,0)`
C. `(-7,0)`
D. None of these

Answer

To solve the problem, we need to find the coordinates of point A on the x-axis where the ray of light reflects. We will follow these steps:

### Step 1: Identify the coordinates of the points involved
- The initial point from which the light ray comes is \( P(1, 2) \).
- The point through which the light ray passes after reflection is \( Q(5, 3) \).
- The point of reflection on the x-axis is \( A(x, 0) \).

### Step 2: Find the mirror image of point Q with respect to the x-axis
The mirror image of point \( Q(5, 3) \) with respect to the x-axis is \( Q'(5, -3) \).

### Step 3: Calculate the slope of the line connecting points P and Q'
The slope \( m \) of the line connecting points \( P(1, 2) \) and \( Q'(5, -3) \) can be calculated using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{-3 - 2}{5 - 1} = \frac{-5}{4} \]

### Step 4: Write the equation of the line PQ'
Using point-slope form of the equation of a line, we can write the equation of line \( PQ' \):
\[ y - y_1 = m(x - x_1) \]
Substituting \( P(1, 2) \) and the slope \( m = -\frac{5}{4} \):
\[ y - 2 = -\frac{5}{4}(x - 1) \]
Simplifying this equation:
\[ y - 2 = -\frac{5}{4}x + \frac{5}{4} \]
\[ y = -\frac{5}{4}x + \frac{5}{4} + 2 \]
\[ y = -\frac{5}{4}x + \frac{5}{4} + \frac{8}{4} \]
\[ y = -\frac{5}{4}x + \frac{13}{4} \]

### Step 5: Find the x-coordinate of point A where y = 0
Since point A lies on the x-axis, we set \( y = 0 \) in the equation:
\[ 0 = -\frac{5}{4}x + \frac{13}{4} \]
Solving for \( x \):
\[ \frac{5}{4}x = \frac{13}{4} \]
\[ x = \frac{13}{5} \]

### Step 6: Write the coordinates of point A
Thus, the coordinates of point A are:
\[ A\left(\frac{13}{5}, 0\right) \]

### Final Answer
The coordinates of point A are \( \left(\frac{13}{5}, 0\right) \).

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