"*Let me see that*!" commanded the Mathematical Ninja, looking at one of the Mathematical Pirate's blog posts. "That's... but that's..."

"It's not wrong!" said the Mathematical Pirate, smugly. "It just *works*!"

"But you're presenting it as *magic*, not as *maths*."

The Mathematical Pirate nodded eagerly. "Lovely magic! How does it work, then, clever-clogs?"

The Mathematical Ninja sighed. "It's not as complicated as all that. You know the circle $C$ is the locus of all of the points a fixed distance $r$ from a given centre $(a, b)$?"

"Terribly sorry, I got bored and stopped listening." He twitched his cutlass to remind the Mathematical Ninja not to try anything.

"In any case, the points where the curve has zero gradient are $(a, b+r)$ and $(a, b-r)$."

"You mean it goes flat at the top and the bottom."

Scowl. "*Which means*, if we differentiate the equation of the circle implicitly..."

"The equation of a circle is $x^2 + y^2 + px + qy + k = 0$" sing-songed the Pirate.

"*An* equation. But yes, differentiated with respect to $x$, you get $2x + 2y \dydx + p + q \dydx = 0$."

"And since $\dydx = 0$, that gives $2x + p = 0$ where the curve is flat *just like I said*."

The Mathematical Ninja ignored him. "You can do the same thing with $\diff xy$ to find where the curve is vertical."

The Mathematical Pirate picked at his fingernails. "Nobody likes a smart-arse, you know. Anyway, what about that radius? Why is that what is is, huh?"

"That, my sea-faring friend, is simple: your equation of the circle tells you where a point is a fixed distance from the centre. Pythagoras says that's $(x-a)^2 + (y-b)^2 - r^2=0$, which is equivalent to your, vastly inferior, form."

"How can it be inferior if they're equivalent, hm?" The cutlass rattled again.

"*Anyway*, sticking $x=a$ and $y=b$ into the left-hand side of this gives you $-r^2$, as required."

"I was once caught in the doldrums for 40 days and 40 nights with only a Daily Mail columnist for company. And even then I wasn't as bored as I am now. I'm going to go and sing a shanty now."

## Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008.
He lives with an espresso pot and nothing to prove.