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Activity 2-5: Conics

www.carom-maths.co.uk. Activity 2-5: Conics. Take a point A, and a line not through A. . Another point B moves so that it is always the same distance from A as it is from the line. . Task: what will the locus of B be? Try to sketch this out. . This looks very much like a parabola ... .

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Activity 2-5: Conics

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  1. www.carom-maths.co.uk Activity 2-5: Conics

  2. Take a point A, and a line not through A. Another point B moves so that it is always the same distance from A as it is from the line.

  3. Task: what will the locus of B be? Try to sketch this out. This looks very much like a parabola...

  4. We can confirm this with coordinate geometry: This is of the form y = ax2 + bx + c, and so is a parabola.

  5. Suppose now we change our starting situation, and say that AB is e times the distance BC,where e is a number greater than 0. What is the locus of B now? We can use a Geogebra file to help us. Geogebra file

  6. We can see the point A, and the starting values for e and q (B is the point (p, q) here). What happens as you vary q? The point B traces out a parabola, as we expect. (Point C traces out the left-hand part of the curve.)

  7. Now we can reduce the value of e to 0.9. What do we expect now? This time the point B traces an ellipse. What would happen if we increased e to 1.1? The point B traces a graph in two parts, called a hyperbola.

  8. Can we get another other curves by changing e? What happens as e gets larger and larger? The curve gets closer and closer to being apair of straight lines. What happens as e gets closer and closer to 0? The ellipse gets closer and closer to being a circle.

  9. So to summarise: This number e is called the eccentricity of the curve. e = 0 – a circle. 0 < e < 1 – an ellipse. e = 1 – a parabola • 1 < e <  –a hyperbola. • e =  –a pair of straight lines.

  10. Now imagine a double cone, like this: If we allow ourselves one plane cut here, what curves can we make? Clearly this will give us a circle.

  11. anda pair of straight lines. This gives you a perfect ellipse… A hyperbola… A parabola… Exactly the same collection of curves that we had with the point-linescenario.

  12. This collection of curves is called ‘the conics’(for obvious reasons). They were well-known to the Greeks – Appollonius (brilliantly) wrote an entire book devoted to the conics. It was he who gave the curves the names we use today.

  13. Task: put the following curve into Autographand vary the constants. How many different curves can you make?ax2 + bxy+ cy2 + ux + vy + w = 0 Exactly the conics and none others!

  14. Notice that we have arrived at three different ways to characterise these curves: 1. Through the point-line scenario, and the idea of eccentricity • 2. Through looking at the curves we can generate • with a plane cut through a double cone • 3. Through considering the Cartesian curves • given by all equations of second degree in x and y. • Are there any other ways to define the conics?

  15. With thanks to:Wikipedia, for helpful words and images. Carom is written by Jonny Griffiths, mail@jonny-griffiths.net

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