The range of g so that we have always a chord of contact of tangents drawn from a real point `(alpha, alpha)` to the circle `x^(2)+y^(2)+2gx+4y+2=0`, is

Options

A. (-3, 0)
B. (-4, 1)
C. (-4, 0)
D. none of these

Explanation

To solve the problem of finding the range of \( g \) such that there is always a chord of contact of tangents drawn from the point \( (\alpha, \alpha) \) to the circle given by the equation \( x^2 + y^2 + 2gx + 4y + 2 = 0 \), we will follow these steps:

### Step 1: Identify the Circle Equation
The equation of the circle is given as:
\[ x^2 + y^2 + 2gx + 4y + 2 = 0 \]
We can rewrite it in the standard form to identify its center and radius.

### Step 2: Determine the Condition for Tangents
For a point \( (x_1, y_1) \) to lie outside a circle, the following condition must hold:
\[ S_1 = x_1^2 + y_1^2 + 2gx_1 + 4y_1 + 2 > 0 \]
In our case, the point is \( (\alpha, \alpha) \).

### Step 3: Substitute the Point into the Condition
Substituting \( x_1 = \alpha \) and \( y_1 = \alpha \) into the condition gives:
\[ \alpha^2 + \alpha^2 + 2g\alpha + 4\alpha + 2 > 0 \]
This simplifies to:
\[ 2\alpha^2 + (2g + 4)\alpha + 2 > 0 \]

### Step 4: Simplify the Inequality
Dividing the entire inequality by 2, we have:
\[ \alpha^2 + (g + 2)\alpha + 1 > 0 \]

### Step 5: Analyze the Quadratic Inequality
The quadratic \( \alpha^2 + (g + 2)\alpha + 1 \) must be greater than zero for all values of \( \alpha \). For a quadratic to be always positive, its discriminant must be less than zero.

### Step 6: Calculate the Discriminant
The discriminant \( D \) of the quadratic \( ax^2 + bx + c \) is given by:
\[ D = b^2 - 4ac \]
Here, \( a = 1 \), \( b = g + 2 \), and \( c = 1 \). Thus, the discriminant is:
\[ D = (g + 2)^2 - 4 \cdot 1 \cdot 1 < 0 \]
This simplifies to:
\[ (g + 2)^2 - 4 < 0 \]

### Step 7: Solve the Inequality
Rearranging the inequality gives:
\[ (g + 2)^2 < 4 \]
Taking the square root of both sides leads to:
\[ -2 < g + 2 < 2 \]
Subtracting 2 from all parts results in:
\[ -4 < g < 0 \]

### Conclusion
Thus, the range of \( g \) such that there is always a chord of contact of tangents from the point \( (\alpha, \alpha) \) to the circle is:
\[ g \in (-4, 0) \]