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)

$(x+y)^{2}-\left(x^{2}+y^{2}\right)=0$
$\Rightarrow x^{2}+y^{2}+2 \,x y-x^{2}-y^{2}=0$
$\Rightarrow 2\, x y=0$
On comparing with $a x^{2}+2 \,h x y+b y^{2}=0$
Here, $a=0, b=0, h=1$
$\therefore \tan \theta=\frac{2 \sqrt{h^{2}-a b}}{a+b}=\infty$
$\Rightarrow \theta=90^{\circ}$
$\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.