$Statement- 1$: If the general equation $x^{2}+y^{2}+2 x y+2 g x+2 f y+4=0$ represents a pair of real lines…
$Statement- 1$: If the general equation $x^{2}+y^{2}+2 x y+2 g x+2 f y+4=0$ represents a pair of real lines then $|\mathrm{g}| \geq 2$.
$Statement- 2$: The equation $a x^{2}+2 h x y+b y^{2}+2 g x+2 f y+c=0$ represents pair
of real lines if $a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}=0$
Statement -1 is false, Statement- 2 is true
Statement -1 is true, Statement- 2 is true; Statement -2 is a correct explanation for Statement-1
Statement -1 is true, Statement- 2 is true; Statement -2 is not a correct explanation for Statement-1
Statement -1 is true, Statement- 2 is false
Solution
The equation represents a pair of lines if $1 \cdot 1 \cdot 4+2 \cdot \mathrm{f} \cdot \mathrm{g} \cdot 1-1 \cdot \mathrm{f}^{2}-1 \cdot \mathrm{g}^{2}-4 \cdot 1^{2}=0$
$\Rightarrow(f-g)^{2}=0 \Rightarrow f=g$
The equation becomes $(x+y)^{2}+2 g(x+y)+4=0$ Which represents pair of parallel lines, which are real provided $(2 \mathrm{~g})^{2}-4 \cdot 4 \cdot 1 \geq 0 \Rightarrow|\mathrm{g}| \geq 2$