Preparing for STEP: Solving a Sphere Tangency Problem

[Positronized]

I'm preparing for the "Sixth-Term Examination Papers" (STEP) this June by going through the past papers. Unfortunately there isn't any solution available for the older past papers AND I've already finished high school which basically means I can't verify my solutions. So I thought it might be a good idea to post some of the problems (which I'm not so sure about) and my worked solution for comments/confirmation here. Will this be acceptable for these forums?

Anyway, here's one to start with:

Consider a sphere of radius a and a plane perpendicular to a unit vector $\hat n$. The centre of the sphere has position vector $\vec d$ and the minimum distance from the origin to the plane is $\ell$. What is the condition for the plane to be tangential to the sphere?

Here's my solution:

There are two planes which have the normal n and is \ell units from the origin. WLOG, consider the plane defined in its point-normal form as the set of all points r such that \left(\mathbf{r}-\ell\mathbf{n}\right)\cdot\mathbf{n}=0 and hence can be simplified to \mathbf{r}\cdot\mathbf{n}=\ell\;\;\mathrm{\left(\star\right)}.
The sphere can be defined as the set of all points r such that \left\Vert\mathbf{r}-\mathbf{d}\right\Vert=a.
If the plane is tangential to the sphere it must be perpendicular to the sphere at the point of tangency. That is, if r is the point of tangency then the direction of radius r-d must be the same as the direction of the plane's normal n. Since \left\Vert\mathbf{r}-\mathbf{d}\right\Vert=a we get \mathbf{r}-\mathbf{d}=\pm a\mathbf{n} since n is a unit vector, hence \mathbf{r}=\mathbf{d}\pm a\mathbf{n}. (the \pm is to take into account both possible directions of n)

Substitute this into (*) above to obtain \left(\mathbf{d}\pm a\mathbf{n}\right)\cdot\mathbf{n}=\ell. Therefore, we get the required condition \ell=\left| \left(\mathbf{n}\cdot\mathbf{d}\right)\pm a \right|.

Is this a correct/complete answer?

 

[Mark44]

Fixed the problem statement to make it readable, and adjusted the formatting.

Seems like an interesting problem that someone might want to take a crack at, despite the age of the post.