21 C EXAMPLES OF MAXIMA AND MINIMA. 



41. Find the length of the shortest normal chord in a para- 



bola, and prove that it intersects the curve nearer the 

 vertex than any other normal chord. 



If 4a be the latus rectum of the parabola the re- 

 quired length is 6a \/3. 



42. Two ships are sailing uniformly with velocities u, v along 



straight lines inclined at an angle 6: shew that if a, b 

 be their distances at one time from the point of inter- 

 section of the courses, the least distance of the ships 



(av bu] sin B 

 is equal to s 



* * 



43. Of all the straight lines drawn from the vertex of a given 

 ellipse to the circumference of the circumscribing circle, 

 determine that for which the portion intercepted be- 

 tween the two curves is a maximum. 



If be the inclination of the straight line to the 

 major axis of the ellipse, and e the eccentricity of the 

 ellipse, 



2e 2 cos s = 3 - e 2 - V{(1 - e 2 ) (9 - e 2 )}. 



44. If an ellipse be described to touch a given semicircle and 

 its diameter symmetrically, its area when a maximum 



will be -j- , r being the radius of the circle. 

 o \ o 



45. An ellipse is inscribed in an isosceles triangle, and has 



one of its axes coincident in direction with the straight 

 line bisecting the vertical angle of the triangle : shew 

 that this axis is two-thirds of the height of the tri- 

 angle when the area of the ellipse is a maximum. 



46. Find what sector must be taken out of a given circle, in 



order that the remainder may form the curved surface 

 of a cone of maximum volume. 



The angle of the sector must be 7r ? ~ ^ . 



y3 



47. Two focal chords are drawn in an ellipse at right angles, 



find when their sum is a maximum, - and when a 

 minimum, 



