Find all points of intersection

[Addez123]

Homework Statement

Find all points where the level surface
$$4x^2+y^2+z^2 = 8$$
and
$$x^2+9y^2=z^2$$
intersects eachother at a 90 degree angle.

Relevant Equations

Surface 1: $$4x^2+y^2+z^2 = 8$$
Surface 2: $$x^2+9y^2 - z^2 = 0$$

First I try to visualize it:
w = Surface 1, is a spheroid
w_2 = Surface 2 is a cone stretching up the z axisThen I calculate their gradients:
$$∇w = (8x, 2y, 2z)$$
$$∇w_2 = (2x, 18y, 2z)$$

The points where they intersect at 90 degrees is when dot product is zero.
$$∇w \cdot ∇w_2 = 0$$
$$16x^2 + 36y^2 - 4z^2 = 0$$
$$z^2 =4x^2 + 8y^2$$
This is a cone stretched differently in x and y axis, but a cone none the less.
Now I need to find where this cone intersects with EITHER the sphere (Surface 1) or the inital cone (Surface 2). Where these intersects, curve 1 and 2 intersect under 90 degree angle. This happens when I set $$z^2 = z_2^2$$.

I use the cone, Surface 2, equation:
$$x^2 + 9y^2 = 4x^2 + 8y^2$$
$$3x^2 - y^2 = 0$$

This is where I get confused. There's no z coordinates so if this was a circle I'd assume it was a cylinder stretching up the whole z-axis.
But the answer is suppose to be either just points, or a curve. It makes no sense that the intersection would create another surface.

What am I doing wrong?

 

[BvU]

Homework Statement:: Find all points where the level curves
...
What am I doing wrong?

My guess is you never make use of this 'level curves' in the problem statement
By the way

is a sphere

No. It is a spheroid.

 

[Addez123]

My guess is you never make use of this 'level curves' in the problem statement
By the way
No. It is a spheroid.

I do use the level surfaces, especially when I combine surface 2 with the surface representing all points at which the two functions form 90 degree angles:

I use the cone, Surface 2, equation:
$$x^2 + 9y^2 = 4x^2 + 8y^2$$

 

[Mark44]

From post 1:

$x^2+9y^2=z^2$

You have a sign error in this equation.

$∇w_2 = (2x, 18y, 2z)$

 

[Addez123]

From post 1:
You have a sign error in this equation.

Thanks! I saw that before tho and changed everywhere but apparently forgot to change that line. Eitherway you can see the dot product is still correct.

 

[Addez123]

Can someone move this to the calculus sub-forum? This isn't homework.

 

[Mark44]

It's enough like a homework question that it should remain here.