So, here’s something cool that Johnny Ball showed me that I hadn’t seen before. I love it when this happens although I fear that as the years roll on and I become a wizened old maths teacher it may happen less and less…
Anyway, here’s a picture to explain, click on it to move the points:
Multiply the moduli of the x-coordinates of the two points where the line intersects the curve and you get the y coordinate where the line intersects the y-axis. (Or another way of looking at it, multiply the x co-ordinates and you get the negative of the y-axis intersect.)
The proof for this is very satisfying and would make a nice extension exercise for Year 12 C1 class. Here are my scribbles. Enjoy!