Advanced book on Mathematics Olympiad

Geometry and Trigonometry 631

0.2

0.6

0.4

0.2

0

-0.2

1

-0.4

-0.6

0 0.4 0.80.6

Figure 80

(r, θ )of the pointL. We haveAM=AL= 2 asinθ 2. In the isosceles triangleLAM,
∠LMA=π 2 −θ 2 ; hence

LM= 2 AMcos

(

π

2

−

θ

2

)

= 2 · 2 asin

θ

2

·sin

θ

2

= 4 asin^2

θ

2

.

Substituting this in the relationOL=OM−LM, we obtain

r=a− 4 asin^2

θ

2

=a[ 1 − 2 ·( 1 −cosθ)].

The equation of the locus is therefore

r=a(2 cosθ− 1 ),

a curve known as Pascal’s snail, or limaçon, whose shape is described in Figure 81.

0 1 2

1.5

1

0.5

0

-0.5

3

-1

-1.5

0.5 1.5 2.5

Figure 81

620.As before, we work with polar coordinates, choosingOas the pole andOAas the
axis. Denote byathe length of the segmentABand byP(r,θ)the projection ofOonto