find partial differential equation of planes having equal x, y intercepts

Question

Find differential equation of all right circular cones having vertex at origin and axis the z-axis.

Answer 1

Equation of plane: f(x,y,z)=ax+by+cz=d; if intercepts are all equal, d/a=d/b=d/c.

Differential equation: ∂f/∂x=a, ∂f/∂y=b, ∂f/∂z=c; if d=abc=(∂f/∂x)(∂f/∂y)(∂f/∂z), bc=ac=ab, so (∂f/∂y)(∂f/∂z)=(∂f/∂x)(∂f/∂z)=(∂f/∂x)(∂f/∂y).

Equation of circular cone: g(x,y,z)=x^2/a^2+y^2/a^2-z^2/b^2=0.

Differential equation: ∂g/∂x=2x/a^2, ∂g/∂y=2y/a^2, ∂g/∂z=-2z/b^2.