If `2(x^(2)+y^(2))+4 lambda x + lambda^(2)=0` represents a circle of meaningful radius, then the range of real values of `lambda`, is

Options

A. R
B. `(0, oo)`
C. `(-oo, 0)`
D. none of these

Explanation

To determine the range of real values of \( \lambda \) such that the equation

\[ 2(x^2 + y^2) + 4\lambda x + \lambda^2 = 0 \]

represents a circle of meaningful radius, we will follow these steps:

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

\[ x^2 + y^2 + 2\lambda x + \frac{\lambda^2}{2} = 0 \]

This is done by dividing the entire equation by 2.

### Step 2: Identify coefficients
In the standard form of a circle, \( (x - h)^2 + (y - k)^2 = r^2 \), we can identify:
- Coefficient of \( x^2 \) is 1
- Coefficient of \( y^2 \) is 1
- Coefficient of \( x \) is \( 2\lambda \)
- Coefficient of \( y \) is 0 (since there is no \( y \) term)
- Constant term \( c = \frac{\lambda^2}{2} \)

### Step 3: Use the condition for a meaningful radius
For the equation to represent a circle of meaningful radius, we need to ensure that:

\[ g^2 + f^2 - c \geq 0 \]

Where:
- \( g = \lambda \)
- \( f = 0 \)
- \( c = \frac{\lambda^2}{2} \)

### Step 4: Substitute and simplify
Substituting the values of \( g \), \( f \), and \( c \) into the condition:

\[ \lambda^2 + 0^2 - \frac{\lambda^2}{2} \geq 0 \]

This simplifies to:

\[ \lambda^2 - \frac{\lambda^2}{2} \geq 0 \]

### Step 5: Further simplification
Combining the terms gives:

\[ \frac{\lambda^2}{2} \geq 0 \]

### Step 6: Analyze the inequality
The inequality \( \frac{\lambda^2}{2} \geq 0 \) holds true for all real values of \( \lambda \). This is because the square of any real number is non-negative.

### Conclusion
Thus, the range of real values of \( \lambda \) such that the equation represents a circle of meaningful radius is:

\[ \lambda \in \mathbb{R} \]