<s>[INST] 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
 [/INST] 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.
</s>