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

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?

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 $$hc = 2fg$$.

• Thanks i got it. Nov 24, 2019 at 1:13

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

Try a few examples

....................

• @Sameernilkhan you did not finish your sentence. Also, instead of telling me what you were thinking, try a few examples. Nov 23, 2019 at 23:44
• I was thinking about conditions like C1) f=0 and c=0 then the equation becomes 2hxy + 2gx = 0 this implies x(hy + g) = 0 i.e. 2 real pair of straight lines. C2) g=0 and c=0 then the equation becomes 2hxy + 2fy = 0 this implies y(hx + f) = 0 i.e. 2 real pair of straight lines. C3) g=0, f=0 and c=0 then the equation becomes 2hxy=0 this implies x=0 and y=0 i.e. 2 real pair of straight lines. Are any more conditions possible? Nov 23, 2019 at 23:59
• I will surely try some examples if you explain the logic behind your condition i. e. c=fg/h. Nov 24, 2019 at 0:02
• I checked the example 6xy + 4x + 12y + 4 = 0. It represents a hyperbola. Nov 24, 2019 at 0:17
• @Sameernilkhan good, I mistakenly left off a factor of $2.$ Let's see, your $h=3$ $g=2$ $f=6$ so $\frac{2fg}{h}= \frac{24}{3} = 8$ Nov 24, 2019 at 0:31