Show that the condition for the pair of lines $ax^2+2hxy+by^2+2gx+2fy+c=0$ to be parallel is $ab=h^2$ and $bg^2=af^2$ or $\dfrac{a}{h}=\dfrac{h}{b}=\dfrac{g}{f}$.

$ab=h^2$ condition is understandable as the acute angle between the lines $\tan\theta=\dfrac{2\sqrt{h^2-ab}}{a+b}$ can be obtained from the corresponding lines going through the origin $ax^2+2hxy+by^2=0$.

In a similar post Deriving conditions for a pair of straight lines to be parallel, it is i think attempted to prove by taking partial derivatives with respect to $x$ and $y$, and taking $𝑎𝑥+ℎ𝑦+𝑔=0$ and $ℎ𝑥+𝑏𝑦+𝑓=0$ to be coincident.

I simply do not understand the logic behind such an attempt ?

And how do I prove that the pair of lines represented by the second order equation are parallel and not coincident ?

Suppose the given equation factors as $$fg=0$$ Partially differentiating by $x$ gives $$f_xg + fg_x=0$$ Partially differentiating by $y$ gives $$f_yg+fg_y = 0$$ Just notice that any point which lies on both $f$ and $g$ also satisfies both above equations.

August 13, 2019 20:02 PM

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