Condition for a pair of straight lines

  • #1
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Main Question or Discussion Point

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.
 

Answers and Replies

  • #2
34,366
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"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.
 
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