Geo3.3

Find the condition that the lines represented by $ax^2+2hxy+by^2=0\,$ are such that the slope of one line is $\lambda\,$ times that of the other.

Let the two lines be $y=mx,y=\lambda mx\,$

We know that $m+\lambda m=\frac{-2h}{b},m(\lambda m)=\frac{a}{b}\,$

$m(1+\lambda)=\frac{-2h}{b},\lambda m^2=\frac{a}{b}\,$

Eliminating m,we get $\lambda[\frac{-2h}{b(1+\lambda)}]^2=\frac{a}{b}\,$

$ab(\lambda+1)^2=4\lambda h^2\,$

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