When does $2hxy + 2gx + 2fy + c =0$, with $h\neq 0$, represent a pair of straight lines?

Question

For the standard conic equation in 2-D plane i.e $$ax^2 + 2hxy + by^2 +2gx +2fy +c =0$$ If $a=b=0$ and $h\neq 0$, then the equation reduces to $$2hxy + 2gx + 2fy + c =0$$

Under what conditions will the above equation represent a pair of straight lines?

Answer 1

when $c = \frac{2fg}{h}$

Try a few examples

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Answer 2

The equation $2hxy + 2gx + 2fy + c =0$ is expected to be factorized as

$$ (px+q)(ty+s)=0$$

in order for it to present two lines. Match the coefficients to get,

$$pt = 2h, \>\>\> qt = 2g,\>\>\> sp=2f, \>\>\> qs = c$$

Then,

$$\frac pq =\frac hg = \frac {2f}c$$

Thus, the condition is $hc = 2fg$.