If the equation ` x^(2)+y^(2)+6x-2y+(lambda^(2)+3lambda+12)=0` represent a circle. Then

Options

A. `lambda in R`
B. `lambda in [1, 2]`
C. `[-2, -1]`
D. none of these

Explanation

To determine the values of \( \lambda \) for which the equation

\[ x^2 + y^2 + 6x - 2y + (\lambda^2 + 3\lambda + 12) = 0 \]

represents a circle, we need to analyze the equation step by step.

### Step 1: Rewrite the equation
The given equation can be rewritten as:

\[ x^2 + y^2 + 6x - 2y + (\lambda^2 + 3\lambda + 12) = 0 \]

### Step 2: Identify the general form of a circle
The general equation of a circle is given by:

\[ x^2 + y^2 + 2gx + 2fy - c = 0 \]

where \( g \) and \( f \) are constants and \( c \) is a positive constant representing the radius squared.

### Step 3: Compare coefficients
From the given equation, we can identify:

- \( 2g = 6 \) → \( g = 3 \)
- \( 2f = -2 \) → \( f = -1 \)
- \( c = -(\lambda^2 + 3\lambda + 12) \)

### Step 4: Use the formula for the radius
The radius \( r \) of the circle is given by:

\[ r = \sqrt{g^2 + f^2 - c} \]

For the equation to represent a circle, the radius must be non-negative:

\[ r \geq 0 \]

### Step 5: Substitute values into the radius formula
Substituting the values of \( g \), \( f \), and \( c \):

\[ r = \sqrt{3^2 + (-1)^2 - (\lambda^2 + 3\lambda + 12)} \geq 0 \]

This simplifies to:

\[ r = \sqrt{9 + 1 - (\lambda^2 + 3\lambda + 12)} \geq 0 \]

### Step 6: Simplify the expression
Simplifying further:

\[ r = \sqrt{10 - (\lambda^2 + 3\lambda + 12)} \geq 0 \]

This leads to:

\[ 10 - \lambda^2 - 3\lambda - 12 \geq 0 \]

which simplifies to:

\[ -\lambda^2 - 3\lambda - 2 \geq 0 \]

### Step 7: Multiply by -1
Multiplying the entire inequality by -1 (and reversing the inequality):

\[ \lambda^2 + 3\lambda + 2 \leq 0 \]

### Step 8: Factor the quadratic
Factoring the quadratic:

\[ (\lambda + 1)(\lambda + 2) \leq 0 \]

### Step 9: Determine the intervals
The roots of the equation are \( \lambda = -1 \) and \( \lambda = -2 \). The inequality \( (\lambda + 1)(\lambda + 2) \leq 0 \) holds true in the interval:

\[ -2 \leq \lambda \leq -1 \]

### Conclusion
Thus, the values of \( \lambda \) for which the equation represents a circle are:

\[ \lambda \in [-2, -1] \]