Advanced book on Mathematics Olympiad

212 4 Geometry and Trigonometry

603.LetA 0 ,A 1 ,...,Anbe the vertices of a regularn-gon inscribed in the unit circle.
Prove that

A 0 A 1 ·A 0 A 2 ···A 0 An− 1 =n.

4.1.3 Conics and Other Curves in the Plane........................

The general equation of a quadratic curve is

ax^2 +by^2 +cxy+dx+ey+f= 0.

Such a curve is called a conic because (except for the degenerate case of two parallel
lines) it can be obtained by sectioning a circular cone by a plane.
The degenerate conics are pairs of (not necessarily distinct) lines, single points, the
entire plane, and the empty set. We ignore them. There are three types of nondegenerate
conics, which up to a change of Cartesian coordinates are described in Figure 27.

082

y

2

0

-4

-6

x

4

6

4

-2

6 10

x

32

y

10-1-2

-1

2

-3

1

0

-2

x

042

-5

-4-6

y 5

10

6

-10

-2^0

y^2 = 4 px x

2

a^2 +

y^2

b^2 =^1

x^2

a^2 −

y^2

b^2 =^1

parabola ellipse hyperbola

Figure 27

The parabola is the locus of the points at equal distance from the point(p, 0 )(focus)
and the linex=−p(directrix). The ellipse is the locus of the points with the sum of
distances to the foci(c, 0 )and(−c, 0 )constant, wherec=

√

|a^2 −b^2 |. The hyperbola
is the locus of the points with the difference of the distances to the foci(c, 0 )and(−c, 0 )
constant, wherec=

√

a^2 +b^2.
Up to an affine change of coordinates, the equations of the parabola, ellipse, and
hyperbola are, respectively,y^2 =x,x^2 +y^2 =1, andx^2 −y^2 =1. Sometimes it is more
convenient to bring the hyperbola into the formxy=1 by choosing its asymptotes as
the coordinate axes.
As conic sections, these curves are obtained by sectioning the circular conez^2 =
x^2 +y^2 by the planesz−x=1 (parabola),z=1 (ellipse), andy=1 (hyperbola).
The vertex of the cone can be thought of as the viewpoint of a person. The projections