These lines have 'Sloppy Slopes'!

Geometry Level 3

$\large ax^2+2hxy+by^2=0$

The above equation represents a pair of straight lines with positive slopes. Given that the slope of one of the line is double the other, then which of the following are necessarily true?

$(i)$ $h<0,b>0,a>0$

$(ii)$ $h<0,b>0,a<0$

$(iii)$ $h>0,b<0,a>0$

$(iv)$ $h>0,b<0,a<0$

This is an original problem and belongs to the set My Creations

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1 solution

Skanda Prasad
Nov 16, 2017

Let the slopes of the two lines be $m$ and $2m$ .

$m+2m=\dfrac{-2h}{b}$

$\implies m=\dfrac{-2h}{3b}$

$\implies$ Either $(h<0,b>0)$ or $(h>0,b<0)$

$2m^2=\dfrac ab$

$\implies m=\sqrt{\frac{a}{2b}}$

$\implies$ Either $(a,b>0)$ or $(a,b<0)$

The value of $m$ has to be positive according to the question.

So if $a,b>0$ then obviously $b>0$ so $h<0$

And if $a,b<0$ then obviously $b<0$ so $h>0$

Hence, $(i)$ and $(iv)$ are the suitable options.

These lines have 'Sloppy Slopes'!

1 solution

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