Frames:

6. Wrap-up

Work through the following problems to check whether you can do everything that the outcomes at the beginning of this unit state.

Solve this problem from the Mathematics HG, Nov 2001 National Paper 2:

2.3 A circle with centre P(x; y) passes through A(4; -1) and touches the line y = 3.  
2.3.1 Determine the equation of the locus of P. (4)
2.3.2 Calculate the gradient of this locus at the point where x = 1. (3)
2.3.3 Determine the equation of the tangent to the locus of P where x = 1. (3)

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You can animate and see the loci above:

  1. shows the locus of the centre of a circle touching a bigger circle.
  2. shows the locus of the centre of a circle touching a bigger circle and passing through a fixed point inside the circle.
  3. shows the locus of the centre of a circle touching a straight line and passing through a fixed point.

Prove that the locus is respectively a circle, an ellipse and a parabola.

Draw the graphs of the following equations by writing them in standard form:

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Resources
David Scher: Geometry in Motion
Xah Lee: Special Plane Curves: Conic Sections
Eric Weisstein: World of Mathematics
Spanish Ministry of Education, Culture and Sport: The Descartes applet