Question

What are represented by the equation $x^2+xy+y^2=0.$

Answer 1

$x^2+xy+y^2=0$

Comparing with the general equation of second degree $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$ ..... $\left(i\right)$ we get

$a=1,b=1,h=\frac{1}{2},f=0,g=0,c=0$

Consider,

$\Delta =abc+2fgh-a{f}^2-b{g}^2-{ch}^2\Delta =0+0-0-0-0=0$

So they represent pair of straight lines

Factorising the given equation with the help of quadratic formula

$x=\frac{-y\pm \sqrt{y^2-4y^2}}{2}x=\frac{-y\pm \sqrt{-3y^2}}{2}$

Clearly it gives a pair of imaginary straight lines.