140 BOOK OF MATHEMATICAL PROBLEMS. 



e is the eccentricity of the hyperbola, which is the locus of the 

 centres of all such conies j prove that 



e* e* 



-x r + -= T- = 4. 



Prove also that the equi-conjugates of the ellipse are parallel to 

 the asymptotes of the hyperbola. 



713. The equation of the conic of minimum eccentricity 

 through four given points is 



x' 



, 

 aa 



' 4*ycoso> y 9 /I 1\ /I 1\ 

 , + - , + YT-, -x( + ,)- y ( T + 77 )+ 1= 0; 

 aa + bb' bb \a a) J \b V J 



the points being on the axes of x, y at distances a, a'; b, b f from 

 the origin. 



714. If <j> be the angle which an axis of any conic through 

 four given points makes with the line bisecting the angle 6 be- 

 tween the axes of the two parabolas through the four points, the 

 eccentricity e of the conic is given by the equation 



e* 4cos'0 



l-e a ~ sin a 0-sm 8 2<* 



715. If 6 be the acute angle between the axes of two para- 

 bolas, the minimum eccentricity of a conic passing through their 



/ 2 



points of intersection is /- . : and either axis of this 



V 1 + sec ' 



conic makes equal angles with the axes of the two parabolas. 



716. Three points A, B, C are taken on an ellipse, the circle 

 about ABC meets the ellipse again in P, and PP' is a diameter ; 

 prove that of all ellipses passing through ABCP' the given ellipse 

 is the one of least eccentricity. 



717. Of all ellipses circumscribing a parallelogram, the one 

 of least eccentricity has its equi-conjugates parallel to the sides of 

 the parallelogram. 



