Math Challenge (again!) #lim_(x->0)(1/x) (or ∞)

Pleae ignore that all-over-the-place title.

 

We're back for another \(MATH\) \(CHALLENGE!\) (in LaTex) I've been experimenting with some of the other features of the... question-maker? Anyway, you can probably see that in my new post about Euler's formula. Anyway anyway, I should tell you the answer to the last one, eh?

\(wxy=10\)

\(wyz=5\)

\(wxz=45\)

\(xyz=12\)

Find \(w+x+y+z\)

 

SOLUTION:

 

Multiply all equations together to get ​\(w^3x^3y^3z^3=2^33^35^3\)

\(wxyz=30\)

Now, divide to get \(w={30\over xyz}\)

Oh, we know \(xyz\)! Substitute to get \(w={5\over2}\)

Repeat to get \(x = 6\), \(y={2\over3}\), and \(z=3\). Add, add, add, and we get

\(w+x+y+z=12{1\over6}\)! Many people got this one.

 

Great! Now for the new problem:

 

Given point \(P\) outside a circle, the shortest distance between \(P\) and the circle's perimeter is 4 units, and the longest distance is 16 units. Find the distance of \(P\) from its tangent.

 

I'm excited to see some answers to this...