<s>[INST] Prove that one of the lines represented by $ax^2+2hxy+by^2=0$ will bisect the angle between the coordinate axes if $(a+b)^2=4h^2$. Prove that one of the lines represented by $ax^2+2hxy+by^2=0$ will bisect the angle between the coordinate axes if $(a+b)^2=4h^2$.

Solution
I calculated the two lines represented by $ax^2+2hxy+by^2=0$ as follows;
here.
$$ax^2+2hxy+by^2=0$$
Multiplying by $a$ on both sides and adding $h^2y^2$ to both sides :
$$ax+hy=\pm y\sqrt{h^2-ab}.$$
What should I do next?
 [/INST] I think this one is easier to work from the other direction.  If $(a+b)^2 = 4 h^2$, then $2 h = \pm (a + b)$.  Thus, the equation for the lines becomes
$$
a x^2 \pm (a + b) xy + b y^2 = 0 \quad \Rightarrow \quad (ax \pm by)(x \pm y) = 0.
$$
Since $(x \pm y)$ is one of the factors, the line $y = \pm x$ will be a solution to the equation.
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