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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:

Here's my solution:

Is this a correct/complete answer?

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 normalnand is [itex]\ell[/itex] units from the origin. WLOG, consider the plane defined in its point-normal form as the set of all pointsrsuch that [itex]\left(\mathbf{r}-\ell\mathbf{n}\right)\cdot\mathbf{n}=0[/itex] and hence can be simplified to [itex]\mathbf{r}\cdot\mathbf{n}=\ell\;\;\mathrm{\left(\star\right)}[/itex].

The sphere can be defined as the set of all points r such that [itex]\left\Vert\mathbf{r}-\mathbf{d}\right\Vert=a[/itex].

If the plane is tangential to the sphere it must be perpendicular to the sphere at the point of tangency. That is, ifris the point of tangency then the direction of radiusr-dmust be the same as the direction of the plane's normaln. Since [itex]\left\Vert\mathbf{r}-\mathbf{d}\right\Vert=a[/itex] we get [itex]\mathbf{r}-\mathbf{d}=\pm a\mathbf{n}[/itex] sincenis a unit vector, hence [itex]\mathbf{r}=\mathbf{d}\pm a\mathbf{n}[/itex]. (the [itex]\pm[/itex] is to take into account both possible directions ofn)

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

Is this a correct/complete answer?

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