Advanced book on Mathematics Olympiad

4.1 Geometry 223

628.Through a pointMon the ellipsoid

x^2

a^2

+

y^2

b^2

+

z^2

c^2

= 1

take planes perpendicular to the axesOx,Oy,Oz. Let the areas of the planar

sections thus obtained beSx,Sy, respectively,Sz. Prove that the sum

aSx+bSy+cSz

is independent ofM.

629.Determine the radius of the largest circle that can lie on the ellipsoid

x^2

a^2

+

y^2

b^2

+

z^2

c^2

= 1 (a>b>c).

630.Leta, b, cbe distinct positive numbers. Prove that through each point of the three-
dimensional space pass three surfaces described by equations of the form

x^2

a^2 −λ

+

y^2

b^2 −λ

+

z^2

c^2 −λ

= 1.

Determine the nature of these surfaces and prove that they are pairwise orthogonal

along their curves of intersection.

631.Show that the equations

x=u+v+w,

y=u^2 +v^2 +w^2 ,

z=u^3 +v^3 +w^3 ,

where the parametersu, v, ware subject to the constraintuvw=1, define a cubic

surface.

We conclude our discussion of coordinate geometry with some problems inndimen-
sions.

Example.Through a fixed point inside ann-dimensional sphere,nmutually perpendicu-
lar chords are drawn. Prove that the sum of the squares of the lengths of the chords does
not depend on their directions.

Solution.We want to prove that the sum in question depends only on the radius of
the sphere and the distance from the fixed point to the center of the sphere. Choose a
coordinate system in which the chords are thenorthogonal axes and the radius of the