Geo3.14

Prove that the pair of lines $(a-\lambda)x^2+2hxy+(b-\lambda)y^2=0\,$ is equally inclined with the pair $ax^2+2hxy+by^2=0\,$

Equation to the pair of angle of bisectors of $(a-\lambda)x^2+2hxy+(b-\lambda)y^2=0\,$ is

$\frac{x^2-y^2}{(a-\lambda)-(b-\lambda)}=\frac{xy}{h}\,$

$\frac{x^2-y^2}{a-b}=\frac{xy}{h}\,$

which is also the equation tot he angle bisectors of $ax^2+2hxy+by^2=0\,$

Hence the given two pairs of lines are equally inclined.