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 \( \lambda \) for which the equation \( 2(x^2 + y^2) + 4\lambda x + \lambda^2 = 0 \) represents a circle with a meaningful radius, we will follow these steps:

### Step 1: Simplify the Equation
We start with the given equation:
\[ 2(x^2 + y^2) + 4\lambda x + \lambda^2 = 0 \]
To simplify, we divide the entire equation by 2:
\[ x^2 + y^2 + 2\lambda x + \frac{\lambda^2}{2} = 0 \]

### Step 2: Compare with the General Form of a Circle
The general equation of a circle is given by:
\[ x^2 + y^2 + 2gx + 2fy + c = 0 \]
From our simplified equation, we can identify:
- \( g = \lambda \)
- \( f = 0 \)
- \( c = \frac{\lambda^2}{2} \)

### Step 3: Find the Radius Condition
The radius \( r \) of the circle can be derived from the formula:
\[ r = \sqrt{g^2 + f^2 - c} \]
Substituting the values we found:
\[ r = \sqrt{\lambda^2 + 0^2 - \frac{\lambda^2}{2}} = \sqrt{\lambda^2 - \frac{\lambda^2}{2}} = \sqrt{\frac{\lambda^2}{2}} \]

### Step 4: Ensure the Radius is Meaningful
For the radius to be meaningful (i.e., \( r \) must be real and non-negative), the expression inside the square root must be greater than zero:
\[ \frac{\lambda^2}{2} > 0 \]
This implies:
\[ \lambda^2 > 0 \]

### Step 5: Determine the Range of \( \lambda \)
The inequality \( \lambda^2 > 0 \) holds true for all \( \lambda \) except \( \lambda = 0 \). Therefore, the range of \( \lambda \) is:
\[ \lambda \in \mathbb{R} \setminus \{0\} \]

### Final Answer
The range of real values of \( \lambda \) for which the equation represents a circle with a meaningful radius is:
\[ \lambda \in (-\infty, 0) \cup (0, \infty) \]

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