Tangent lines,Sphere and Cone problem

[Elena Baby]

Hello,this is the problem:
Prove that every point on lines passing through the point (0,0,3) that are tangent to a sphere with O(0,0,0) as the centre and radus=1,are on a cone.

Sorry if the problem seems off.I had to translate it from my own language.

This is my attempt:
The standard equation for a cone is:x^2/a^2 +y^2/b^2 =z^2/c^2 .
The given sphere is:x^2+y^2+z^2=1.
We can write the sphere equation as:x^2+y^2=-(z^2-1).

I need to have a vector function so that I can make it into a cone equation(?)

I hope it's the right way.I'd be really happy if you'd consider giving me a hint to how should I start.

 

[Dr.Peterson]

The problem as you state it makes sense. But your equation for a cone assumes the vertex is the origin, which is not true here.

It might help to know the context. You mention vectors; can you just write a vector-based proof, or must you use the sort of equations you've shown? Without knowing what sort of proofs you are learning, it's hard to say how you should start.

 

[Elena Baby]

The problem as you state it makes sense. But your equation for a cone assumes the vertex is the origin, which is not true here.

It might help to know the context. You mention vectors; can you just write a vector-based proof, or must you use the sort of equations you've shown? Without knowing what sort of proofs you are learning, it's hard to say how you should start.

I'm pretty sure I should usethe equations,because my professor hadn't taught anything about multivariable functions and vectors when he gave us this problem as homework.

 

[Dr.Peterson]

I myself would tend to use straight geometry. But if your problem came specifically in the context of these equations, then you can start by writing the correct equation for a cone with vertex at (0,0,3), not (0,0,0), and with its axis along the z-axis.

Then you need to decide how what you have learned applies to tangent lines to a sphere. Given a line through (0,0,3), how would you determine whether it is tangent?

If I were helping you face to face (and knew your language, whatever it is), I would be looking through your textbook to see what you have learned, particularly about tangency. In your course, would you do this using calculus methods, or the fact that a tangent line is perpendicular to the radius, or using the negative-reciprocal slope, or something else?

The more you tell us about the context of the problem, particularly what you have most recently learned, the better we can help.