Question

Show that the equation $b{x}^3-2hxy+a{y}^2=0$ represents a pair of straight lines which are at right angles to the pair given by the equation $a{x}^2+2hxy+b{y}^2=0$

Answer 1

$b{x}^2-2hxy+a{y}^2=0$

Let $y=mx$ be the of line reprented by the given equation

$b{x}^2-2hm{x}^2+a{m}^2{x}^2=0b-2hm+a{m}^2=\mathrm{0.....}\left(i\right)$

For perpendicular line the slope of the line will be $\frac{-1}{m}$

Using it in $\left(i\right)$

$b-2h\left(\frac{-1}{m}\right)+a{\left(\frac{-1}{m}\right)}^2=0b{m}^2+2hm+a=\mathrm{0.....}\left(ii\right)$

Equation of line represented by $\left(ii\right)$ is

$a{x}^2+2hxy+b{y}^2=0$

Hence proved.