Find the equations of all planes that pass through the points (1,1,1) and (2,0,1) and also are tangent to the surface $x^2 + y^2 + z^2 = 1$

Question

Find the equations of all planes that pass through the points (1,1,1) and (2,0,1) and also are tangent to the surface $x^2 + y^2 + z^2 = 1$

Answer 1

EDIT: Misread the question.

For our purposes it is useful to view this sphere as the level surface (three-dimensional analogue of the contour lines for functions of two variables) of the function $f:x,y,z\mapsto x^{2}+y^{2}+z^{2}$. This is because the gradient $\nabla f(a,b,c)$ of $f$ at a point $(a,b,c)$ is normal to the level surface $\{ (x,y,z) \in\mathbf{R}^{3} : f(x,y,z) = f(a,b,c) \}$ at the point $(a,b,c)$, a fact we can use to build our tangent plane: for example, since

$$ \nabla f (1,1,1) = \begin{bmatrix}2\\2\\2\end{bmatrix} $$

we have that

$$ 2(x-1) + 2(y-1) + 2(z-1) = 0 $$

is the equation of the plane tangent to the to the sphere at one of your points $(1,1,1)$.

This may help as well