Ask Uncle Colin: Tangents to a circle | Colin

Dear Uncle Colin,

I’m told that two lines through $(0,12)$ are tangent to the circle with equation $(x-6)^2 + (y-5)^2 = 17$ and I need to find their equations – but I’m getting in a muddle. Can you help?

– Terribly Awkward Numbers, Getting Equations Not Trivial

Hi, TANGENT, and thank you for your message!

This certainly is a non-trivial one, and the numbers are definitely awkward!

The main difficulty for most people here is keeping a clear distinction between variables – such as the $x$ and $y$ that represent points in space – and parameters – such as $m$, which represents the…

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The Evolution of Polynomials | Colin

It’s always fascinating to see what’s going on in textbooks of the olden days, and National Treasure @mathsjem recently found a beauty of its type. Look at those whences! Check out the subjunctives! It thrills the heart, doesn’t it?1

What caught my attention, though, was evolution – in this context, taking square or cube (or, presumably, higher) roots of an expression.

I know! There’s a whole chapter on each of them. Looks complicated… pic.twitter.com/DWPH4e96Ip

— Jo Morgan (@mathsjem) January 3, 2018

I haven’t studied this method in detail – it looks similar in flavour to finding…

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Ask Uncle Colin: A Troublesome Triangle | Colin

Dear Uncle Colin,

I couldn’t make head nor tail of this geometry problem: “If $a:b=12:7$, $c=3$, and $B\hat{A}C = 2 B\hat{C}A$, find the length of the sides $a$ and $b$.”

– Totally Rubbish In Geometry

Hi, TRIG, and thank you for your message! (And don’t put yourself down like that, it doesn’t help.)

I would probably start by letting $a=12x$ and $b=7x$, and seeing what came out of the equations. (I’d also let angle $B\hat{A}C=2\alpha$ and $B\hat{C}A=\alpha$, meaning $A\hat{B}C=\pi – 3\alpha$ – if I can avoid using that, I will.)
Let’s start with the sine rule

We have…

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Using Units to Deal With Density | Colin

Glancing over sample papers for the new GCSE, I stumbled on this:

Zahra mixes 150g of metal A and 150g of metal B to make 300g of an alloy.

Metal A has a density of $19.3 \unit{g/cm^3}$.

Metal B has a density of $8.9 \unit{g/cm^3}$.

Work out the density of the alloy.

I don’t think I’m being mean to say that this would stump the majority of students. It’s probably designed to.
Ahoy there, Mathematical Pirate!

“Aharr! I spies a compound unit!”

“Are you going to talk in that ridiculous manner for the whole blog post? I happen know you’re from Windsor.”

“The rough end of…

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Ask Uncle Colin: A modulus power | Colin

Dear Uncle Colin,

How would you work out $9^{41} \pmod{61}$?

– Funky Exponential Result, Missed A Tutorial

Hi, FERMAT, and thanks for your question!

I think the answer is “ponderously”! There are only 61 possible answers (in fact, 60, because you know 61 is not a factor of $9^{41}$).

I’d start by playing around a bit: $9^2$ is 81, which is congruent to $20 \pmod{61}$.

Carrying on, $9^3 = 180 \equiv (-3) \pmod{61}$

That’s interesting – we now have a small number! Squaring it gives us $9^6 \equiv 9 \pmod{61}$, and we can check that $9^5 \equiv 1 \pmod{61}$ (in fact, it’s…

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Sprinkle on the sugar, eat the lot | Colin

A Christmas Pudding Puzzle

I swear, this one came up in real life!

My partner made a Christmas pudding for the most recent festive season. Delicious, it was.

When it was about half-eaten, I went to microwave a portion. “Hang on,” she said: “there might be a coin in there.”

As is traditional, she had mixed in at least one threepenny bit1; as is habitual, she couldn’t remember whether it was one coin or half a dozen.

We could remember that her brother was the only one so far to have found a coin.

So, how likely is it that my portion, let’s say a sixth of the remainder, contains…

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Ask Uncle Colin: A Misbehaving Inverse | Colin

Dear Uncle Colin,

Why is $\arcsin\br{\sin\br{\frac {6}{7}\pi}}$ not $\frac{6}{7}\pi$?

– A Reasonable Conclusion Seems Incorrect Numerically

Hi, ARCSIN, and thanks for your message!

On the face of it, it does seem like a reasonable conclusion: surely feeding the output of $\sin(x)$ into its inverse function should get you back where you started?

There’s a wrinkle, though: $\arcsin(x)$ is not exactly the inverse function of $\sin(x)$.
Defining a function

A function is typically defined in three parts: it has:
A name – here, $\sin(x)$
A definition – for example, $\lim_{n\to…

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