Prove that every line normal to the cone with equation z=sqrt(x^2+y^2) intersects the z-axis.

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- Oct 10th 2010, 07:17 AMesum0209Prove
Prove that every line normal to the cone with equation z=sqrt(x^2+y^2) intersects the z-axis.

- Oct 10th 2010, 09:01 AMtonio

Define $\displaystyle F(x,y,z):=\sqrt{x^2+y^2}-z\Longrightarrow$ the normal line to this surface at any given point $\displaystyle (x_0,y_0,z_0)$ is given by the parametric equations

$\displaystyle x(t):=x_0+\frac{x_0}{\sqrt{x_0^2+y_0^2}}t\,,\,\,y( t)=y_0+\frac{y_0}{\sqrt{x_0^2+y_0^2}}t\,,\,\,z(t)= z_0-t\,,\,\,t\in\mathbb{R}$ .

Now prove that there exists $\displaystyle t_1\in\mathbb{R}\,\,s.t.\,\, x(t_1)=y(t_1)=0$ ...

Tonio - Oct 10th 2010, 10:48 AMesum0209
- Oct 10th 2010, 11:28 AMPlato
What happens if $\displaystyle t = \dfrac{{ - 1}}

{{\sqrt {x_0 ^2 + y_0 ^2 } }}?$ - Oct 10th 2010, 11:28 AMtonio