Question

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 $|g| \geq 2$.
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 $a b c+2 f g h-a f^{2}-b g^{2}-c h^{2}=0$

• A. Statement -1 is false, Statement- 2 is true
• B. Statement -1 is true, Statement- 2 is true; Statement -2 is a correct explanation for Statement-1
• C. Statement -1 is true, Statement- 2 is true; Statement -2 is not a correct explanation for Statement-1
• D. Statement -1 is true, Statement- 2 is false

Answer 1

Answer: (D)

The equation represents a pair of lines if
$1 \cdot 1 \cdot 4+2 \cdot f \cdot g \cdot 1-1 \cdot f^{2}-1 \cdot g^{2}-4 \cdot 1^{2}=0$
$\Rightarrow (f-g)^{2}=0 \Rightarrow f=g$
The equation becomes $(x+y)^{2}+2 g(x+y)+4=0$
Which represents pair of parallel lines,
which are real provided $(2 g)^{2}-4 \cdot 4 \cdot 1 \geq 0 $
$\Rightarrow |g| \geq 2$