Question

Statement-1 : If the general equation $x^2+y^2+2xy+2gx+2fy+4=0$ represents a pair of real lines then $|g|\ge 2$.
Statement- 2 : The equation
$a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ represents pair of real lines if $abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

• 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 $(x+y{)}^2+2g(x+y)+4=0$
Which represents pair of parallel lines,
which are real provided $(2g{)}^2-4\cdot 4\cdot 1\ge 0$
$\Rightarrow |g|\ge 2$