Question

The equation ${(x+y)}^2-(x^2+y^2)=0$ represents:

• A. a circle
• B. two lines
• C. two parallel lines
• D. two mutually perpendicular lines

Answer 1

Answer: (D)

$(x+y{)}^2-\left(x^2+y^2\right)=0$
$\Rightarrow x^2+y^2+2xy-x^2-y^2=0$
$\Rightarrow 2xy=0$
On comparing with $a{x}^2+2hxy+b{y}^2=0$
Here, $a=0,b=0,h=1$
$\therefore \tan \theta =\frac{2\sqrt{h^2-ab}}{a+b}=\infty$
$\Rightarrow \theta =90^{\circ }$
$\therefore$ The given equation represents two mutually perpendicular lines.
Note: In second degree homogeneous equation if $a+b=0$, then the pair of lines are perpendicular.