This brief post owes its inspiration to @MrNiksMathClass who shared some of the below on twitter recently. So we all know that y=x^{2} produces the most exquisitely beautiful and proportioned curve, rivalled perhaps only by the catenary of y = coshx.

I was delighted to recently find this interesting property of this most marvellous parabola.

- Select two points on the curve y=x
^{2}, each either side of the y axis
- Draw a straight line through points
- The y intercept of the straight line is equal to the negative product of the x coordinates of the two original points

Of course, I’m sure you may be reaching for a pencil now to prove it. My approach was to label the x coordinates of the two original points a and b and the intersect c. I then found the gradient of the line between them and the equation of that line. Then, I substituted in a (or b) as x and out popped c = -ab as required. A student of mine used a slightly different approach by using the expression for the intersection between the straight line and the curve and multiplying the two solutions to find that they resulted in –c.

It’s perhaps even nicer to start with the curve y=-x^{2} as this results in the y intercept being equal to the product of the x coordinates of the two original points. Again, this is fairly straight forward to prove, but make sure you adjust for the new gradient on your line.

This problem is a lovely vehicle for exploring coordinate geometry and quadratics. Geogebra works well as a vehicle to test out some ideas if you wanted to leave room for the students to explore the relationship of the y intercept before you give them the result. Then they can switch to pen and paper to prove it.

Some possible extensions of this task include:

- What happens if we start with y=kx
^{2} or y=-kx^{2}. Always good to generalise and it works out quite smoothly in this case.
- Where does the line between the two coordinate points cross the x axis? Expressing this in terms of a and b is interesting and the case of when a=b can be explained both geometrically and algebraically.
- What is the ratio of the length of the line segments ac and bc? This one’s on my to do list!

Any comments or ideas greatly appreciated.

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You also might mention that the y-coordinate of C is the geometric mean of the y-coordinates of A and B

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