Advanced book on Mathematics Olympiad

4.1 Geometry 217

A few algebraic computations yield

2 y 0 t+t^2 (x−x 0 )=a(x+x 0 )+b.

This shows thatxis a rational function of the slopet. The same is true fory.Astvaries,

(x, y)describes the whole conic. This is a rational parametrization of the conic, giving

rise to Euler’s substitutions. In their most general form, Euler’s substitutions are

√

ax^2 +bx+c−y 0 =t(x−x 0 ).

They are used for rationalizing integrals of the form

∫

R(x,

√

ax^2 +bx+c)dx,

whereRis a two-variable rational function.

614.Compute the integral

∫

dx

a+bcosx+csinx

,

wherea, b, care real numbers, not all equal to zero.

615.Consider the system

x+y=z+u,

2 xy=zu.

Find the greatest value of the real constantmsuch thatm≤ xyfor any positive

integer solution(x,y,z,u)of the system, withx≥y.

We conclude our incursion into two-dimensional geometry with an overview of var-

ious famous planar curves. The first answers a question of G.W. Leibniz.

Example.What is the path of an object dragged by a string of constant length when the

end of the string not joined to the object moves along a straight line?

Solution.Assume that the object is dragged by a string of length 1, that its initial co-

ordinates are( 0 , 1 ), and that it is dragged by a vehicle moving along thex-axis in the

positive direction. Observe that the slope of the tangent to the curve at a point(x, y)

points toward the vehicle, while the distance to the vehicle is always equal to 1. These

two facts can be combined in the differential equation

dy

dx

=−

y

√

1 −y^2

.