
Tangent Planes
I am trying to find all planes that are tangent to both x^2 + y^2 + z^2 = 1 and x^2 + y^2 + 2z = 0.
I tried using gradient to get the general equation of the planes for each but after that I got stuck.
Any suggestions would be greatly appreciated. Thanks!

Here's one way.
The general tangent plane to the sphere at the point is , and the general tangent plane to the paraboloid at is .
If these planes are identical, we must have .
So , and . Using we deduce that .
But . Therefore and implying .
Hence , and .
To summarise: with the common tangent planes are where , touching the sphere at and the paraboloid at .

(1)
(2)
Use a parametric equation to replace (2) ,
And the normal of the paraboloid ( in vector form , but not unit vector) is :
And the tangent plane of it is :
OR
Since the required planes touch the unit sphere , the distance of the plane from the oringin is 1 .
Solving the equation , gives
Finally , substitute back to the equation of the plane ,
the required plane is obtained :
In fact , there are infinite planes which satisify your requirement