Mad props to Dave Radcliffe

Today I learned a really neat (and new to me) basic algebraic technique from Dave Radcliffe on twitter.

This morning James Tanton posed this interesting problem:

Do the pts of intersection of two parabolas y=ax^2+bx+c and x=dy^2+ey+f always lie on common circle?

— James Tanton (@jamestanton) December 18, 2014

I started thinking about it, but work got in the way. Checking back a little later I saw an amazing solution (that fits in a tweet!) from Dave Radcliffe:

@jamestanton Yes! Solve the equations for x^2 and y^2 respectively, then add the equations to obtain the equation of a circle. Nice problem!

— David Radcliffe (@daveinstpaul) December 18, 2014

Super clever. If you want a circle, you’ll need an equation like and the conditions of the problem essentially allow you to force that equation!

For fun I plugged these three equations into Wolfram Alpha to take a peek at the solution:

(1) ,

(2) , and

(3)

The third equation comes from applying Dave’s idea to the first two equations. Sure enough, you get this picture:

The Wolfram Alpha code is here:

Wolfram Alpha code for drawing the above picture

Not every day you learn a new high school algebra technique from a tweet – thanks for posting the cool solution, Dave!

[Post publishing note]

Two great Desmos programs help give you a feel for the problem. First from Chris Lusto:

Chris Lusto’s Desmos Program

and second from Justin Lanier:

Justin Lanier’s Desmos Program