Answer to Question #314194 in Calculus for Amorah

Question

Please help me with this question. Consider the surfaces in R^3 defined by the equations f(x,y)= 2 sqrt(x^2 + y^2) and g(x,y)= 1 + x^2 + y^2. (a) what shapes are described by f,g and their intersection?. (b) Give a parametric equation describing the intersection

Answer 1

(a) $z=f\left( x,y \right) =2\sqrt{x^2+y^2}$ describes a cone, $z=g\left( x,y \right) =1+x^2+y^2$ describes a parabaloid.

The intersection:

$\left\{ \begin{array}{c} z=2\sqrt{x^2+y^2}\\ z=1+x^2+y^2\\\end{array} \right. \Rightarrow \left\{ \begin{array}{c} z=2\sqrt{x^2+y^2}\\ 2\sqrt{x^2+y^2}=1+x^2+y^2\\\end{array} \right. \Rightarrow \left\{ \begin{array}{c} z=2\sqrt{x^2+y^2}\\ \left( \sqrt{x^2+y^2}-1 \right) ^2=0\\\end{array} \right. \Rightarrow \\\Rightarrow \left\{ \begin{array}{c} x^2+y^2=1\\ z=2\\\end{array} \right.$

This is a circle.

(b)

$x=\cos \varphi ,y=\sin \varphi ,z=2,0\leqslant \varphi \leqslant 2\pi$