If was trying to do the following question.
Prove that the pair of straight lines $ax^2+ 2hxy + by^2 + 2gx +2fy+ c = 0$ are parallel if $h^2 = ab$ and $bg^2 = af^2$
Here's what I tried.
I partially differentiated the give equation with respect to x and y to get these equations
$ax + hy + g = 0$ and
$hx + by + f = 0$
Now since intersection of these lines gives the intersection of the given pair of straight lines, both of these must be parallel to have no intersection. Therefore,
$a/h = h/b = g/f$
I am doubtful if my method is correct. Please provide with better solution if there's any. Thanks.