A ray of light coming from origin after reflectiion at the point `P (x ,y)` of any curve becomes parallel to x-axis, the , equation of the curve may be :

Options

A. `y^2 =x`
B. `y^2 =2x +1`
C. `y^2 = 4x `
D. `y^2= 4x +1`

Answer

To solve the problem, we need to find the equation of a curve such that a ray of light coming from the origin reflects off a point \( P(x, y) \) on the curve and becomes parallel to the x-axis.

### Step-by-Step Solution:

1. **Understanding the Reflection**:
- A ray of light coming from the origin (0, 0) reflects off a point \( P(x, y) \) on the curve. After reflection, the ray becomes parallel to the x-axis. This means that the angle of incidence is equal to the angle of reflection.

2. **Identifying the Geometry**:
- The ray from the origin to the point \( P(x, y) \) can be represented as a line with a slope of \( \frac{y}{x} \).
- After reflection, the ray must have a slope of 0 (since it is parallel to the x-axis).

3. **Using the Reflection Law**:
- The angle of incidence \( \theta \) can be described as \( \tan^{-1}(\frac{y}{x}) \).
- The angle of reflection must equal the angle of incidence, which means we can use the property of the tangent to find the relationship between the slope before and after reflection.

4. **Finding the Curve**:
- The curve that satisfies this reflection property is a parabola. The general form of a parabola that opens to the right is given by:
\[ y^2 = 4ax \]
- Here, \( a \) is a constant that determines the width of the parabola.

5. **Determining the Focus**:
- The focus of the parabola \( y^2 = 4ax \) is at the point \( (a, 0) \).
- Since the ray of light comes from the origin and reflects at point \( P(x, y) \), we can set the focus at the origin for the ray to reflect correctly.

6. **Final Equation**:
- To satisfy the condition that the ray reflects off the curve and becomes parallel to the x-axis, we can conclude that the equation of the curve is:
\[ y^2 = 4ax \]
- This indicates that the curve is indeed a parabola.

### Conclusion:
The equation of the curve such that a ray of light coming from the origin reflects at a point \( P(x, y) \) and becomes parallel to the x-axis is given by:
\[ y^2 = 4ax \]