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Thread: Prove

  1. #1
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    Prove

    Prove that every line normal to the cone with equation z=sqrt(x^2+y^2) intersects the z-axis.
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  2. #2
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    Quote Originally Posted by esum0209 View Post
    Prove that every line normal to the cone with equation z=sqrt(x^2+y^2) intersects the z-axis.

    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
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  3. #3
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    Quote Originally Posted by tonio View Post
    Now prove that there exists $\displaystyle t_1\in\mathbb{R}\,\,s.t.\,\, x(t_1)=y(t_1)=0$ ...

    Tonio
    Sorry, but I don't know to prove this... Can u help me a bit more?

    Thanks in advance
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  4. #4
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    What happens if $\displaystyle t = \dfrac{{ - 1}}
    {{\sqrt {x_0 ^2 + y_0 ^2 } }}?$
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  5. #5
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    Quote Originally Posted by esum0209 View Post
    Sorry, but I don't know to prove this... Can u help me a bit more?

    Thanks in advance

    No. If you're dealing with partial derivatives you must be able to cope with this: show that the solution to $\displaystyle x(t_1)=0$ is the same as the solution to $\displaystyle y(t_1)=0$ . This is HS stuff.

    Tonio
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