First slide

Definition of a circle

Question

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

Moderate

A

$λ \in R$
B

$λ \in [1, 2]$
C

$[- 2, - 1]$
D

none of these

Solution

The given equation will represent a circle if
$9 + 1 - (λ^{2} + 3 λ + 12) \geq 0 [∵ g^{2} + f^{2} - c \geq 0]$
$\begin{array}{l} \Rightarrow - λ^{2} - 3 λ - 2 \geq 0 \\ \Rightarrow λ^{2} + 3 λ + 2 \leq 0 \Rightarrow (λ + 1) (λ + 2) \leq 0 \Rightarrow λ \in [- 2, - 1] \end{array}$

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