How many parameters does the set of all spheres, which satisfy the given condition, depend on?

Question

How many parameters does the set of all spheres, which satisfy the given condition, depend on?

(i) Spheres that pass through the given point.

(ii) Spheres that touch the given line

(iii) Spheres that touch the given plane

In general, the sphere given by $$(x-a)^2+(y-b)^2+(z-c)^2=R^2$$ depends on $a,b,c$ and $R$.

For (i), the spheres depend on only $R$ because as we know the point we can find all centers of all spheres. So, I thought $\textbf{one parameter}$.

But I'm not sure about my thoughts neither do I have some thoughts for (ii) and (iii). Any help is appreciated.

Answer 1

I agree with @Ross Millikan, and i have posted similar proofs for other 2 cases .

Answer 2

It is not true that there is only one parameter for i. The number of parameters is the number of variables you must set to pick one sphere out of the collection. Your general equation for the sphere shows that a sphere without restriction depends on four parameters, $a,b,c,R$. If you fix one point on the sphere, you can still pick the center freely. Having picked the center, $R$ is fixed as the distance to the fixed point, so the collection of spheres going through a point has three parameters. For ii and iii presumably touch means be tangent to. I find question iii to be easier to visualize, so would suggest starting there.