Find equations for all the planes that intersect the $y$-axis at $y = 1$ and the $z$-axis at $z = 2$, and are tangent to the sphere $$( − 2)^2 + ^2 + ^2 = 4.$$ Do not use calculus.
So basically we have the sphere $$( − 2)^2 + ^2 + ^2 = 4$$ and I'm looking for tangent planes that intersect the points $(0,1,0)$ and $(0,0,2)$ and that will create a line and im looking for planes that are tangent to the sphere and contain both those points.
So far I only have the $x=0$ plane because of inspection and I'm not sure about the rest of the planes whether there are only a few more planes or if there is a region on the sphere where the tangent planes contain the line. If so I cant seem to find the range of the values because I'm stuck figuring out how to express a point on the plane as a general point of the sphere.
Not really sure what to do here I've spent 5 hours racking my brain trying to do this.
Hint:
the planes passing through the given points have equation $$ax+y+\frac{1}{2}z=1$$ and are tangent to the given sphere if the distance from the center of the sphere is the radius of the sphere.