Question

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

• A. R
• B. $(0,+\infty )$
• C. $(-\infty ,0)$
• D. none of these

Answer 1

The correct option is D none of these

Given equation of circle is
$2(x^2+y^2)+4\lambda x+{\lambda }^2=0$
$x^2+y^2+2\lambda x+\frac{{\lambda }^2}{2}=0$
$\Rightarrow (x+\lambda {)}^2+y^2=\frac{{\lambda }^2}{2}$
Center is at $(-\lambda ,0)$
Radius is $\frac{|\lambda |}{\sqrt{2}}$
For a circle of meaningful radius,
$r>0$
$\Rightarrow \frac{|\lambda |}{\sqrt{2}}>0$
$\Rightarrow \lambda \in R-\left\{0\right\}$
Hence, range of $\lambda$ is $(-\infty ,0)\cup (0,\infty )$