Question

If the equation $x^2+y^2+6x-2y+({\lambda }^2+3\lambda +12)=0$ represent a circle, then

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

Answer 1

The correct option is B $\lambda \in [1,2]$

The general equation of circle is :

$x^2+y^2+2gx+2fy+c=0$, where

Center = $(-g,-f)$ and

Radius = $\sqrt{g^2+f^2-c}$

For the given equation to represent circle, radius must be real

$\to \sqrt{(-3{)}^2+(1{)}^2-({\lambda }^2+3\lambda +12)}\in R$

$\to 10-({\lambda }^2+3\lambda +12)\ge 0$

$\to {\lambda }^2+3\lambda +2\le 0$

$\to (\lambda +2)(\lambda +1)\le 0$

$\to \lambda \in [1,2]$

Hence, (B) is the correct answer.