Question

3
2. The vertex of a right circular cone is (1, 0, 1), and
its axis is equally inclined to the three coordinate
axes. If the point (1, 1, 1) lies on the cone, find
its equation​

Answer 1

Answer:

This is a great example of why it’s useful to understand other coordinate systems. Cones are simple in cylindrical coordinates. We should slightly modify the usual cylindrical framework to allow the cone’s axis to be the x-axis rather than z. Here’s the relationship between my cylindrical system and the Cartesian system.

Let r=y2+z2−−−−−−√ and θ=atan2(y,z) .

I’ll put the tip of the cone at the origin and make its slope m>0 . I will allow its radius and length to grow without bound, but restricting the value of x will make it a finite cone if that’s your preference.

We know that the radius grows at a constant rate m with respect to x . That’s what the slope — the arbitrary positive constant m — tells us. So r=mx . That’s it. That’s the equation of your cone