Parametrisation of a curve(intersection of a circular cone and a plane)

Question

I am trying to find a parametrisation of the intersection of the graphs of the functions: $f(x, y) = \sqrt{x^2+y^2}$ and $g(x, y) = 20 + x − y$. I used a graphing tool, which gave me the following result: I tried $x=cost, y=sint, z=20+cost-sint$ but that does not work. I would appreciate any help.

Answer 1

$$f(x, y) = \sqrt{x^2+y^2}$$ and $$g(x, y) = 20 + x − y$$ intersect where $$ \sqrt{x^2+y^2}=20 + x − y$$

Square both sides to get $$x^2+y^2=400 +x^2 + y^2 +40x-40y -2xy$$

$$40x-40y -2xy=-400$$

Solve for $y$ to get $$y=\frac {40x+400}{40+2x}$$

Parametrize: $$ x=t$$

$$y=\frac {40t+400}{40+2t}$$

$$z=\sqrt {t^2+(\frac {40t+400}{40+2t})^2} $$