What are the equations of the three straight lines represented by $x^3+bx^2y+cxy^2+y^3=0$ when $b+c=-2$?

Question

I am given that $$x^3 + bx^2y + cxy^2 + y^3 = 0$$ represents three straight lines if $b + c = -2$.

Is there a way to find the equations of the three lines separately?

I tried factorizing the equation but wasn't able to get anywhere. Is there a way to get the separate equations using partial differentiation?

Answer 1

If $b+c=-2$, you can write $b=-k$ and $c=k-2$, so

$$\begin{align}x^3+bx^2y+cxy^2+y^3&=x^3-kx^2y+(k-2)xy^2+y^3\\&=x^3-kx^2y+kxy^2-2xy^2+y^3\\&=(x^3-xy^2)-(kx^2y-kxy^2)-(xy^2-y^3)\\&=(x-y)(x(x+y)-kxy-y^2)\\&=(x-y)(x^2-(k-1)xy-y^2)\end{align}$$

You can factor the latter part by completing the square but the computation is a bit ugly.

The three lines will be:

$$x=y\\ x=y\left(\frac{k-1}2+\sqrt{1+\left(\frac{k-1}2\right)^2}\right)\\ x=y\left(\frac{k-1}2-\sqrt{1+\left(\frac{k-1}2\right)^2}\right)$$

where $k$ is just $-b$, a constant, so the entire expression in the brackets is a constant, the reciprocal of their slopes.

Answer 2

Observe that all three straight lines pass through the origin $(0, 0)$

Let the three straight lines are

$y = m_1x$, $y = m_2x$ and $y = m_3x$

Then $$ (y -m_1x)(y -m_2x)(y -m_3x) \equiv x^3 + bx^2y + cxy^2 + y^3$$

Comparing the coefficients, we have

$m_1 m_2 m_3 = -1$

$m_1 m_2 + m_1 m_3 + m_2 m_3 = b$

$m_1 + m_2 + m_3 = -c$

These are the three roots of the equation $m^3 + cm^2 + bm +1 = 0$

If $b + c = -2$, then $m=1$ must be a root of the above equation. The other quadratic factor is $m^2 - bm - m - 1$ and has the roots $\frac{1}{2} (1 + b \pm \sqrt{b^2 + 2 b + 5})$

Finally the three straight lines are

$y = x$

$y = \frac{x}{2} (1 + b + \sqrt{b^2 + 2 b + 5})$

and

$y = \frac{x}{2} (1 + b - \sqrt{b^2 + 2 b + 5})$