A ray of light moving parallel to x-axis gets reflected from a parabolic mirror whose equation is `4( x+ y) - y^(2) = 0` . After reflection the ray pass through the `pt(a, b)`. Then the value of a + b is

Options

A. 2
B. 1
C. -2
D. -1

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript while providing a clear and structured solution.

### Step-by-Step Solution

1. **Identify the Equation of the Parabola**:
The given equation of the parabola is:
\[ 4x + y - y^2 = 0 \]
We can rearrange this equation to express it in a standard form.

2. **Rearranging the Equation**:
Rearranging the equation gives:
\[ y^2 - 4y = 4x \]
Now, we will complete the square on the left side.

3. **Completing the Square**:
To complete the square for \(y^2 - 4y\), we add and subtract \(4\):
\[ y^2 - 4y + 4 - 4 = 4x \]
This simplifies to:
\[ (y - 2)^2 = 4x + 4 \]
or
\[ (y - 2)^2 = 4(x + 1) \]

4. **Identifying the Vertex and Focus**:
From the equation \((y - 2)^2 = 4(x + 1)\), we can identify:
- Vertex: \((-1, 2)\)
- The value of \(A\) (distance from the vertex to the focus) is \(1\) (since \(4A = 4\)).

5. **Finding the Focus**:
The focus of the parabola is located at:
\[ (x + A, y) = (-1 + 1, 2) = (0, 2) \]

6. **Ray of Light Reflection**:
A ray of light moving parallel to the x-axis will reflect off the parabolic mirror and pass through the focus. Thus, after reflection, the ray passes through the point \((0, 2)\).

7. **Identifying Point (a, b)**:
From the problem, we know that the ray passes through the point \((a, b)\). Since we found the focus to be \((0, 2)\), we can set:
\[ a = 0 \quad \text{and} \quad b = 2 \]

8. **Calculating \(a + b\)**:
Finally, we calculate:
\[ a + b = 0 + 2 = 2 \]

### Final Answer
The value of \(a + b\) is:
\[ \boxed{2} \]