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Exploring Conics with Geogebra: Parabola, Ellipse, Hyperbola, Circle

Investigate the loci of points moving relative to a line and a point, varying eccentricity values to form conic sections. Discover the properties of parabolas, ellipses, hyperbolas, and circles using geometric and algebraic methods. Play with Autograph to explore different curves described by second-degree equations. Learn about the ancient Greek mathematician Apollonius and his work on conics.

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Exploring Conics with Geogebra: Parabola, Ellipse, Hyperbola, Circle

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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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