Question

The equation x^2+y^2-2xy+2x++1=0 represents a parabola

Answer 1

$\\ x^{2}+y^{2}-2xy+2x+1=0$ is the given equation.

The general equation of 2nd degree in two variables x,y is

$ax^{2}+by^{2}+2hxy+2gx+2fy+c=0$

Comparing this with the given equation we get a=1,b=1,h=-1,g=1,f=0,c=1.

$\Delta=\begin{vmatrix} a & h & g\\ h & b & f\\ g & f & c \end{vmatrix}$ =$\begin{vmatrix} 1 & -1 & 1 \\ -1 & 1 & 0\\ 1 & 0 & 1 \end{vmatrix}$ =1+1*(-1)+1*(-1)=-1$\neq$ 0.

$h^{2}-ab=1-1=0$ .

Therefore it represents a parabola.