Analytic geometry straight line problem Prove that two straight lines represented by the equation $x^3+y^3+bx^2y+cxy^2=0$ will be at right angles if $b+c=-2$
I didn't know that even straight lines like planes can be represented by a combined equation. can someone please explain how this happens and how to find the individual lines so that the angle between them may be determined. 
Thanks

Suppose you have two lines through the origin, one of slope $\alpha$ and the other of slope $\beta$:
\begin{align*}
l_1:\quad y &= \alpha x & \alpha x - y &= 0 \\
l_2:\quad y &= \beta x & \beta x - y &= 0
\end{align*}
Then you can multiply (powers of) these euqations to obtain a combined equation, which will be zero if and only if one of its factor polynomials is zero:
\begin{align*}
l_1^2 \cdot l_2&: &
\alpha^{2} \beta x^{3}
- (\alpha^{2} + 2 \, \alpha \beta) x^{2} y
+ (\beta + 2 \, \alpha) x y^{2}
- y^{3}
&= 0\\
l_1 \cdot l_2^2&: &
\alpha \beta^{2} x^{3}
- (\beta^{2} + 2 \, \alpha \beta) x^{2} y
+ (\alpha + 2 \, \beta) x y^{2}
- y^{3}
&= 0
\end{align*}
But these equations don't have the required form. It is enough to concentrate on the first equation, since $l_1$ and $l_2$ are structurally equivalent, so you might as well swap them. To turn $l_1^2\cdot l_2$ into the required form, you should multiply it with $-1$ in order to get $+y^3$ instead of $-y^3$. Then you get a monomial of $+x^3$ exactly if $\alpha^2\beta=-1$. So we now want
\begin{align*}
\alpha^2\beta &= -1 &
\\
\alpha^2 + 2\alpha\beta &= b &
\\
\beta + 2\alpha &= -c
\\
b + c &= -2
\end{align*}
Typing this in e.g. Wolfram Alpha you will get one real and two complex solutions:
\begin{align*}
b &= -1 & b &= -1-2i & b &= -1+2i \\
c &= -1 & c &= -1+2i & c &= -1-2i \\
\alpha &= 1 & \alpha &= -i & \alpha &= i \\
\beta &= -1 & \beta &= 1 & \beta &= 1 \\
\alpha\beta &= -1 & \alpha\beta &= -i & \alpha\beta &= i
\end{align*}
The real solution satisifes your requirement: the slopes multiply to $-1$, so the lines are orthogonal. Moreover, there is only a single pair of lines with that property. The complex solutions don't satisfy your requirement, but you'll probably want to exclude these from your consideration in any case.
