Solving for Points on a Sphere w/ Parallel Plane

[stripes]

Homework Statement

Find the points on the sphere x² + y² + z² = 1 where the tangent plane is parallel to the plane 2x + y - 3z = 2.

Homework Equations

z - z₀ = f_x(x₀, y₀)(x - x₀) + f_y(x₀, y₀)(y - y₀₎)

That's all I believe

The Attempt at a Solution

This chapter in our text does not require the use of vectors (eg dot and cross product), so I need to do this strictly algebraically. I found both first order partials, but didn't really know what to do after that. I plug in these partials to the expression above, but I don't think it's possible to solve for x₀ and y₀ explicitly.

If someone could guide me in the right direction, it would be much appreciated. Thank you all in advance!

 

[lanedance]

first find the normal to the plane

then note any vector from the origin through, will be orthogonal to the sphere face where it intersects it