Advanced book on Mathematics Olympiad

630 Geometry and Trigonometry

01 0.2

40

0.8

20

(^0) 0.6
-20
0.4
-40
Figure 79
with the equation of the cardioid, we find the possible answers to the problem as( 0 , 0 ),
( 2 , 0 ),(−^14 ,
√ 3
4 ), and(−
1
4 ,−
√ 3
4 ). Of these the origin has to be ruled out, since there the
cardioid has a corner, while the other three are indeed points where the tangent to the
cardioid is vertical.
617.LetAB =a and consider a system of polar coordinates with poleAand axis
AB. The equation of the curve traced byMis obtained as follows. We haveAM=r,
AD=cosaθ, andAC=acosθ. The equalityAM=AD−ACyields the equation
r=
a
cosθ
−acosθ.
The equation of the locus is thereforer = asin
(^2) θ
cosθ. This curve is called the cisoid of
Diocles (Figure 80).
618.LetObe the center andathe radius of the circle, and letMbe the point on the
circle. Choose a system of polar coordinates withMthe pole andMOthe axis. For
an arbitrary tangent, letIbe its intersection withMO,Tthe tangency point, andPthe
projection ofMonto the tangent. Then

OI=

OT

cosθ

=

a

cosθ

.

Hence

MP=r=(MO+OI)cosθ=

(

a+

a

cosθ

)

cosθ.

We obtainr=a( 1 +cosθ), which is the equation of a cardioid (Figure 80).

619.Working with polar coordinates we place the pole atOand axisOA. Denote
byathe radius of the circle. We want to find the relation between the polar coordinates