x2+y2−2xy+2x+1=0 is the given equation.
The general equation of 2nd degree in two variables x,y is
ax2+by2+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.
Δ=∣∣ahghbfgfc∣∣ =∣∣1−11−110101∣∣ =1+1*(-1)+1*(-1)=-1= 0.
h2−ab=1−1=0 .
Therefore it represents a parabola.
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