A light ray emerging from the point source placed at `P(2,3)` is reflected at a point `Q` on the y-axis. It then passes through the point `R(5,10)dot` The coordinates of `Q` are (a) `(0,3)` (b) `(0,2)` (c) `(0,5)` (d) none of these

Options

A. (0,3)
B. (0,2)
C. (0,5)
D. none of these

Explanation

To solve the problem, we need to find the coordinates of point \( Q \) on the y-axis where a light ray reflects. The light ray originates from point \( P(2, 3) \) and passes through point \( R(5, 10) \) after reflecting at point \( Q \).

### Step-by-Step Solution:

1. **Identify the Coordinates of Point Q**:
Since point \( Q \) is on the y-axis, its x-coordinate is \( 0 \). We can denote the coordinates of \( Q \) as \( Q(0, a) \), where \( a \) is the unknown y-coordinate we need to find.

2. **Use the Concept of Reflection**:
The angle of incidence is equal to the angle of reflection. This means we can set up a relationship between the slopes of the lines \( PQ \) and \( QR \).

3. **Calculate the Slope of Line PQ**:
The slope of line \( PQ \) can be calculated using the coordinates of points \( P(2, 3) \) and \( Q(0, a) \):
\[ \text{slope of } PQ = \frac{a - 3}{0 - 2} = \frac{a - 3}{-2} \]

4. **Calculate the Slope of Line QR**:
The slope of line \( QR \) can be calculated using the coordinates of points \( Q(0, a) \) and \( R(5, 10) \):
\[ \text{slope of } QR = \frac{10 - a}{5 - 0} = \frac{10 - a}{5} \]

5. **Set the Slopes Equal**:
Since the angle of incidence equals the angle of reflection, we can set the absolute values of the slopes equal to each other:
\[ \left| \frac{a - 3}{-2} \right| = \left| \frac{10 - a}{5} \right| \]
This leads to two cases to consider:
- Case 1: \( \frac{a - 3}{-2} = \frac{10 - a}{5} \)
- Case 2: \( \frac{a - 3}{-2} = -\frac{10 - a}{5} \)

6. **Solve Case 1**:
\[ 5(a - 3) = -2(10 - a) \]
\[ 5a - 15 = -20 + 2a \]
\[ 5a - 2a = -20 + 15 \]
\[ 3a = -5 \implies a = \frac{-5}{3} \quad \text{(not valid as } a \text{ must be positive)} \]

7. **Solve Case 2**:
\[ 5(a - 3) = 2(10 - a) \]
\[ 5a - 15 = 20 - 2a \]
\[ 5a + 2a = 20 + 15 \]
\[ 7a = 35 \implies a = 5 \]

8. **Final Coordinates of Point Q**:
Thus, the coordinates of point \( Q \) are \( (0, 5) \).

### Conclusion:
The coordinates of point \( Q \) are \( (0, 5) \), which corresponds to option (c).