Condition for a pair of straight lines

[rajeshmarndi]

While determining the condition for the pair of straight line equation

$ax^2+2hxy+by^2+2gx+2fy+c=0$

or $ax2+2(hy+g)x+(by^2+2fy+c)=0$ (quadratic in x)

$x = \frac{-2(hy+g)}{2a} ± \frac{√((hy+g)^2-a(by^2+2fy+c))}{2a}$

The terms inside square root need to be a perfect square and it is in quadratic in y i.e $√[(h^2-ab)y^2+2(hg-af)y+(g^2-ac)]$
or $√[Ay^2+By+C] = √[(√Ay+\frac{B}{2√A})^2-\frac{B^2-4AC}{2A}]$.
Hence the condition is taken as $B^2-4AC=0$.

My question even if $B^2-4AC≠0$, the term inside square root can still be a perfect square e.g $4^2-7=3^2, 6^2-11=5^2,23^2-45=22^2$ and so on...

Thank you.

 

[mfb]

"Perfect square" doesn't refer to integers here, it means the square root has to simplify to m*y+c for some m,c. It does so only if you can write the expression under the root as (m*y+c)², which means the absolute term that is left over has to be zero.