Q.Determine whether $x^2-4xy+4y^2=0$ represents two distinct lines or two coincident lines.

Comparing the equation with $ax^2+2hxy+by^2=0$ and checking the sign of $h^2-ab$ tells us immediately whether the two lines are distinct or the same line repeated.

Comparing $x^2-4xy+4y^2=0$ with the general homogeneous form $ax^2+2hxy+by^2=0$: $a=1$, $2h=-4 \Rightarrow h=-2$, $b=4$.

Discriminant check. $h^2-ab = (-2)^2 - (1)(4) = 4-4=0$. Since $h^2=ab$, the pair of lines is coincident — the equation represents one line, counted twice.

Independent check (via factoring). $x^2-4xy+4y^2$ should then be a perfect square. Indeed:

$x^2-4xy+4y^2 = (x-2y)^2$ …

Mixing up the coincidence condition ($h^2=ab$) with the perpendicularity condition ($a+b=0$) is a common confusion — here $a+b=1+4=5\ne0$, so the lines (even though coincident) are not perpen …