200 BOOK OF MATHEMATICAL PROBLEMS. 



950. The distances of any point from two fixed points are r, r, 

 and a maximum or minimum value off(r, r') for points lying on 

 a given curve is c ; prove that the curve f(r, r') = c will touch 

 the given curve. 



951. In the straight line bisecting the angle A of a tri- 

 angle ABC is taken a point P ; prove that the difference of 

 the angles APB t APC will be a maximum when AP is a mean 

 proportional between AB and AC. 



952. Find the maximum and minimum values of a normal 

 chord of a given ellipse : proving that if one exist other than the 



axes, the eccentricity must be > j^ , and that the length of such 



v^ 



a chord will be -- 3 , where 2, 2b are the axes. 



953. Normals are drawn to an ellipse at the extremities of 



two conjugate diameters : prove that a maximum or minimum 







distance of their point of intersection from the centre is -~ 



o 

 provided the eccentricity is > p . Exa'mine which of the two it is. 



954. If x + y + z = 3c, f(x) . f(y) . f(z) will be a maximum or 

 minimum when x = y = z = c, according as 



955. The minimum area which can be included between two 

 parabolas, whose axes are parallel and at a distance c, and 



which cut each other at right angles in two points, is c 8 -^-~ 



95 G. is the centre of curvature at a point P of a given 

 ellipse, and OP*, OQ are normals drawn from : prove that, 

 if a 2 < 26*, P'Q' has its minimum value when the eccentric angle 



