Show that the Plane $P:ax+by+cz=0$ cuts the Cone $C:xy+yz+zx=0$ in perpendicular lines if $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=0$ The solution starts as: Since the plane passes though the vertex, $$\frac{x}{a}=\frac{y}{b}=\frac{z}{c}$$ is a generator of the cone. I am not able to see (visualise) why the Normal of the given plane is a generator of the cone. I can see that the plane passes through the vertex.
(Further the solution says since the dr's of the line satisfy cone equation, we plug in $a,b,c$ in C to get $ab+bc+ca=0$ and thus the required result; I am okay with this part)
 A: The equation of the cone can be written
$$(x+y+z)^2-(x^2+y^2+z^2)=0$$
Or under a form using a dot product
$$\underbrace{\frac{1}{\sqrt{x^2+y^2+z^2}}\begin{pmatrix}x\\y\\z\end{pmatrix}}_{\text{directing vector of a generatrix}} \ . \ \underbrace{\begin{pmatrix}1\\1\\1\end{pmatrix}}_U=\pm 1 \tag{1}$$
(sign + for the half cone containing $U$, sign - for the other one).
This explains that the cone is circular with axis directed by $U=(1,1,1)$.
Let us denote by $D$ the vector $(a,b,c)$, that we can assume with unit norm WLOG.
(1) is verified in particular for points of type $(x,0,0)$ which means that the $x$ axis is a generator line of the cone. Same thing for the $y$ and the $z$ axes. If we rotate them around $U$, we generate the cone in its totality.
Let $e_x,e_y,e_z$ be the 3 unit vectors of these coordinate axes.
If the 2 intersection lines of the plane with the cone have orthogonal unit norm directing vectors $V_1,V_2$, this pair can be brought by a certain rotation $R$ around $U$ onto the pair $(e_1,e_2)$.
As $D=(a,b,c)$ is orthogonal to $V_1$ and to $V_2$ (this is what plane equation $ax+by+cz=0$ expresses), rotation $R$ sends the line directed by $D=(a,b,c)$ onto the line directed by $e_3$ [just because cross product $D=\pm V_1 \times V_2$ is sent onto $e_3=\pm e_1 \times e_2$]. This means that the line directed by $D$ belongs to the cone, otherwise said
$$\dfrac{x}{a}=\dfrac{y}{b}=\dfrac{z}{c}$$
is a generator line of the cone.
If it can help you to visualize a little better the situation, see the picture below with the (half) cone as seen from behind as a kind of asian conical hat with orthogonal (red) axes defined by $(e_x,e_y,e_z)$, and $(V_1,V_2,D)$ represented by green axes.

