Preparing for STEP: Solving a Sphere Tangency Problem

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In summary, a sphere tangency problem involves finding the point(s) where a sphere and another geometric shape touch or intersect. It is important to learn how to solve these problems as they are commonly encountered in various fields of science and engineering. The steps involved in preparing for solving a sphere tangency problem include understanding the problem, identifying the given information, sketching the problem, and applying relevant mathematical concepts and formulas. Some common strategies for solving sphere tangency problems include using the Pythagorean theorem, the distance formula, and the equation of a circle, as well as breaking the problem down into smaller parts. It is also helpful to double-check calculations and practice similar problems to improve understanding and problem-solving skills.
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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:
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 [itex]\ell[/itex] units from the origin. WLOG, consider the plane defined in its point-normal form as the set of all points r such 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, 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 [itex]\left\Vert\mathbf{r}-\mathbf{d}\right\Vert=a[/itex] we get [itex]\mathbf{r}-\mathbf{d}=\pm a\mathbf{n}[/itex] since n is 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 of n)

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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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.
 

Related to Preparing for STEP: Solving a Sphere Tangency Problem

1. What is a sphere tangency problem?

A sphere tangency problem involves finding the point(s) where a sphere and another geometric shape (such as a plane, line, or another sphere) touch or intersect.

2. Why is it important to learn how to solve sphere tangency problems?

Sphere tangency problems are commonly encountered in various fields of science and engineering, such as in geometry, physics, and computer graphics. Learning how to solve these problems can help us better understand the relationships between different geometric shapes and improve our problem-solving skills.

3. What are the steps involved in preparing for solving a sphere tangency problem?

The steps involved in preparing for solving a sphere tangency problem include understanding the problem, identifying the given information, sketching the problem, and applying relevant mathematical concepts and formulas to find the solution.

4. What are some common strategies for solving sphere tangency problems?

Some common strategies for solving sphere tangency problems include using the Pythagorean theorem, the distance formula, and the equation of a circle. It is also helpful to visualize the problem and break it down into smaller, more manageable parts.

5. Are there any specific tips or tricks for solving sphere tangency problems?

One helpful tip is to always double-check your calculations and make sure you are using the correct formulas. It can also be useful to work through similar practice problems to improve your understanding and problem-solving skills.

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