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 first rewrite the equation of the parabola, identify its properties, and then find the coordinates of the focus. Finally, we will calculate \( a + b \).

### Step 1: Rewrite the equation of the parabola
The given equation of the parabola is:
\[ 4(x + y) - y^2 = 0 \]
We can rearrange this equation to isolate \( y^2 \):
\[ y^2 = 4(x + y) \]
This simplifies to:
\[ y^2 - 4y = 4x \]

### Step 2: Complete the square
To complete the square for the \( y \) terms, we add 4 to both sides:
\[ y^2 - 4y + 4 = 4x + 4 \]
This can be factored as:
\[ (y - 2)^2 = 4(x + 1) \]

### Step 3: Identify the vertex and focus
The equation \((y - 2)^2 = 4(x + 1)\) is in the standard form of a parabola that opens to the right:
\[ (y - k)^2 = 4p(x - h) \]
where \((h, k)\) is the vertex and \( p \) is the distance from the vertex to the focus.

From our equation:
- The vertex \((h, k) = (-1, 2)\)
- The value of \( 4p = 4 \) implies \( p = 1 \)

Thus, the focus is located at:
\[ (h + p, k) = (-1 + 1, 2) = (0, 2) \]

### Step 4: Find the point through which the ray passes
According to the properties of parabolas, any ray of light parallel to the axis of symmetry (which is the x-axis in this case) will pass through the focus after reflection. Therefore, the ray of light will pass through the point \((0, 2)\).

### Step 5: Calculate \( a + b \)
Given that the point through which the ray passes is \((a, b) = (0, 2)\), we can find:
\[ a + b = 0 + 2 = 2 \]

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