(i) What type of conic is the conic x2 − 4xy + 4y2 − 4x = 0?
(ii) What type of conic is the conic x2 − 4xy + 4y2 − 4 = 0?
The type depends only on the second order terms Since those both have the same second order terms, they are the same type. For $\displaystyle Ax^2+ Bxy+ Cx^2$, look at $\displaystyle 4AC- B^2$. If that is positive you have an ellipse or circle. If it is negative, a hyperbola, and if it is 0, a parabola.