history of mathematics - National STEM Centre

Descartes
Activity 8.9 Normals
Figure 8.2
108
Figure 8.2 shows an ellipse with its major axis along the jc-axis. The length of the
major axis is 4, and the length of the minor axis is 2.
1 Check that the equation of this ellipse is x 2 -4x + 4y 2 = 0.
The point P(p, 0) is on the x-axis, and the jc-coordinate of the point S on the ellipse,
is 3. The circle with centre P and radius PS touches the ellipse.
2 Show that
a the equation of the circle is y 2 = -x 2 +2px + 9-| — 6p
b the x-coordinate of the point of contact S of the circle and the ellipse is
x 2 + }(4 -8p)x + 8p -13 = 0
c this implies that | (4 - 8/7) = -6 and 8/7-13 = 9.
3 Find the equation of the normal at S to the ellipse.
4 Find in the same way the equation of the normal at (4,3) to the parabola
Descartes, Fermat and analytical geometry
Descartes and his La Geometric are often closely associated with the development
of analytical or coordinate geometry, but this is only partly valid. La Geometric is
not a book on analytical geometry, but, as you have seen, it acted as a starting point
for the study of analytical geometry. But Descartes was not the only one who did the
groundwork; among others who contributed was his contemporary, Pierre Fermat.
Fermat's mathematical work was partly in the same field as that of Descartes. There
are some similarities, but also a number of clear differences. In this section you will
briefly compare some aspects of their work.
Analytical geometry is the use of coordinate systems and algebraic methods in
geometry. In a coordinate system, a point is represented as a set of numbers
(coordinates) and a curve is represented by an equation for a specific set of points.
So the geometrical properties of curves and figures can be investigated with the help
of algebra.
Activity 8.10 Reflecting on Descartes, 26
1 Describe at least one aspect of La Geometric closely related to the above
definition of analytical geometry.
2 Describe one aspect that does not relate to this definition.