# Visualisation of plane cutting cone in perpendicular lines

I came across a problem which requires to prove that plane $ax + by + cz=0$ cuts cone $xy+yz+xz = 0$ in perpendicular lines if $1/a + 1/b + 1/c = 0$

Solution to the problem says that since given cone is generated by three mutually perpendicular planes, hence plane $ax + by + cz=0$ will cut it in perpendicular lines if normal to plane through vertex (0,0,0) lies on cone itself.

I am unable to visualise graphically how such a plane can cut cone in perpendicular lines. Why is it necessary for normal to plane through $(0,0,0)$ to lie on cone?

I am assuming that lines being referred in question are the boundaries of the cone which plane would touch when cutting across cone. Any graph/picture would be thankful.

• This cone is the surface of rotation of any of the coordinate axes about the line $x=y=z$. So in particular, each of the coordinate planes intersects this cone in a pair of perpendicular lines: two of the coordinate axes.
– amd
Commented Sep 15, 2017 at 21:09
• Is it possible to visualise this somewhere? Commented Sep 18, 2017 at 8:50
• You could plot the cone and play around with various planes through the origin in Wolfram|Alpha or a tool like GeoGebra.
– amd
Commented Sep 18, 2017 at 9:19