The range of values of `alpha` for which the line `2y=gx+alpha` is a normal to the circle `x^2=y^2+2gx+2gy-2=0` for all values of `g` is

Options

A. `[1,oo)`
B. `[-1,oo)`
C. `(0,1)`
D. `(- oo,1]`

Explanation

To find the range of values of \( \alpha \) for which the line \( 2y = gx + \alpha \) is normal to the circle given by the equation \( x^2 = y^2 + 2gx + 2gy - 2 = 0 \), we can follow these steps:

### Step 1: Rewrite the Circle Equation
First, we rewrite the circle equation in a more standard form. The given equation is:
\[ x^2 - y^2 - 2gx - 2gy + 2 = 0 \]
This can be rearranged to:
\[ x^2 - 2gx + y^2 + 2gy - 2 = 0 \]

### Step 2: Identify the Center of the Circle
From the standard form of the circle, we can identify the center of the circle. The center \( (h, k) \) can be found using the coefficients of \( x \) and \( y \):
\[ h = g, \quad k = -g \]
So, the center of the circle is \( (g, -g) \).

### Step 3: Determine the Condition for Normality
For the line \( 2y = gx + \alpha \) to be normal to the circle, it must pass through the center of the circle. Therefore, we substitute the coordinates of the center into the line equation:
\[ 2(-g) = g(g) + \alpha \]
This simplifies to:
\[ -2g = g^2 + \alpha \]
Rearranging gives us:
\[ \alpha = -2g - g^2 \]

### Step 4: Analyze the Expression for \( \alpha \)
The expression for \( \alpha \) can be rewritten as:
\[ \alpha = -(g^2 + 2g) \]
This is a downward-opening parabola in terms of \( g \).

### Step 5: Find the Range of \( \alpha \)
To find the maximum value of \( \alpha \), we need to find the vertex of the parabola \( g^2 + 2g \). The vertex \( g \) of a parabola \( ax^2 + bx + c \) is given by:
\[ g = -\frac{b}{2a} = -\frac{2}{2 \cdot 1} = -1 \]
Substituting \( g = -1 \) back into the expression for \( \alpha \):
\[ \alpha = -((-1)^2 + 2(-1)) = -1 + 2 = 1 \]
Thus, the maximum value of \( \alpha \) is \( 1 \).

### Step 6: Determine the Range of \( \alpha \)
Since the parabola opens downwards, \( \alpha \) can take any value less than or equal to \( 1 \). Therefore, the range of values for \( \alpha \) is:
\[ (-\infty, 1] \]

### Final Answer
The range of values of \( \alpha \) for which the line \( 2y = gx + \alpha \) is normal to the circle for all values of \( g \) is:
\[ (-\infty, 1] \]