Geo3.13

Prove that one of the lines $ax^2+2hxy+by^2=0\,$ will bisect the angle between the coordinate axes if $(a+b)^2=4h^2\,$

Equation to the pair of axes is $xy=0\,$

Equation to the bisectors of the angle between the axes are $x^2-y^2=0,y=\pm x\,$

This line belongs to $ax^2+2hxy+by^2=0\,$

$ax^2+2hx(\pm x)+bx^2=0\,$

$(a+b)=\pm 2h\,$

Squaring on both sides, we get

$(a+b)^2=4h^2\,$