202 BOOK OF MATHEMATICAL PROBLEMS. 



962. If u, x, y, z be the distances of any point from four 

 given points not in one plane, and f(u, x, y y z) be a maximum 

 or minimum; 



1 - a* - 6* - c* + 2abc \du) ~ 1 - a* - b'" - c' J 



/ 1 - a* - b" - c 9 + 2a'b'c 



a, 6, c, a', i', c denoting the cosines of the angles between the 

 distances (y, z), (, a), (, y), (M, a?), (M, y), (u, z) respectively. 



9 63. P, P' are contiguous points of a curve, PO, P'O are 

 drawn at right angles to. the radii vectores SP, SP' ; prove that 



the limiting value of PO, as P' moves up to P, is -^ . 



964. S t II are two fixed points, P is a point moving so that 

 the rectangle SP, HP is constant : prove that straight lines 

 drawn from S, H at right angles respectively to SP, HP will 

 meet the tangent at P to the locus of P in points equidistant 

 from P. 



965. In the curve y 3 = 3ax*x 3 , the tangent at P meets 

 the curve again in Q ; prove that 



tan QOx + 2 tan POx = 0, 



being the origin. Prove also that, if the tangent at P be ft 

 normal at Q, P lies on the curve 



966. In the curve y* = a*x, the greatest acute angle between 



q 



two tangents which intersect on the curve is tan" 1 -r . 



967. A tangent to the curve x^ + y* = a* which makes an 

 angle tan" 1 * with the axis of x is also a normal to the 



