For which values of $k$ the equation represents an ellipse

Question

For which values of $k$ the following equation represents an ellipse: $$kx(x+2) + 2y(x-2y+1) + 1 = 0?$$

Answer 1

We have: $$kx^2+2xy-4y^2+2kx+2y+1=0$$ The discriminant of the equation is $$2^2-4(-4)k=4+16k$$ $$4+16k<0\iff k<-\frac14$$ i.e. it represents an ellipse when $k<-\frac14$.