Consider the curved mirror `y =f(x)` passing through `(0, 6)` having the property that all light rays emerging from origin, after getting reflected from the mirror becomes parallel to x-axis, then the equation of curve, is

Options

A. `y^(2)=4(x-y) or y^(2)=36(9+x)`
B. `y^(2)=4(1-x) or y^(2)=36(9-x)`
C. `y^(2)=4(1+x) or y^(2)=36(9-x)`
D. None of these

Answer

To solve the problem, we need to find the equation of a curved mirror represented by the function \( y = f(x) \) that passes through the point \( (0, 6) \) and reflects light rays from the origin such that they become parallel to the x-axis.

### Step-by-Step Solution:

1. **Understanding the Problem**:
The light rays coming from the origin (0,0) reflect off the mirror and become parallel to the x-axis. This means that the focus of the parabola (the point from which the light rays emanate) must be at the origin.

2. **Identifying the Type of Parabola**:
Since the light rays are parallel to the x-axis after reflection, the parabola must open horizontally. The standard form of a horizontally opening parabola is given by:
\[ (y - k)^2 = 4p(x - h) \]
where \((h, k)\) is the vertex and \(p\) is the distance from the vertex to the focus.

3. **Setting the Focus and Vertex**:
Given that the focus is at the origin (0,0) and the parabola passes through the point (0,6), we can set the vertex of the parabola at some point \((h, k)\). Since the focus is at (0,0), we can assume that the vertex is at \((h, k)\) where \(k\) is the y-coordinate of the vertex.

4. **Using the Point (0,6)**:
The parabola passes through the point (0,6). We can substitute this point into the parabola's equation to find the relationship between \(h\), \(k\), and \(p\).

5. **Equation of the Parabola**:
Since the focus is at (0,0), we can assume the vertex is at (h, k) where \(k = 6\) (since the parabola passes through (0,6)). The equation becomes:
\[ (y - 6)^2 = 4p(x - h) \]

6. **Finding the Value of \(h\)**:
Since the focus is at (0,0), we know that the distance from the vertex to the focus is \(p\). Thus, we have:
\[ p = h \]
Therefore, we can rewrite the equation as:
\[ (y - 6)^2 = 4h(x - h) \]

7. **Finding the Specific Parabola**:
We need to find the correct parabola from the given options. We will check each option to see if it passes through (0,6) and has the focus at (0,0).

- **Option A**: \(y^2 = 4(x - 6)\)
- Substituting (0,6): \(36 = 4(0 - 6)\) → \(36 \neq -24\) (not valid)

- **Option B**: \(y^2 = 4(1 - x)\)
- Substituting (0,6): \(36 = 4(1 - 0)\) → \(36 \neq 4\) (not valid)

- **Option C**: \(y^2 = 4(1 + x)\)
- Substituting (0,6): \(36 = 4(1 + 0)\) → \(36 \neq 4\) (not valid)

8. **Conclusion**:
After checking the options, none of them satisfy the conditions. Therefore, we conclude that the equation of the curve is not among the provided options.