JEXAMPLES OF MAXIMA AND MINIMA. 215 



34. Having given an angle of a quadrilateral and the two 



opposite sides, prove that the area will be 9, maxi- 

 mum when the given angle is equidistant from the 

 other angles. 



It follows from the preceding Example that the two 

 sides which contain the given angle must he equal in 

 order to ensure a maximum area ; for if they were not 

 equal the area of the quadrilateral would be increased 

 by changing these two sides into two equal sides. 



35. Find the least ellipse which can be described about a 



given parallelogram, and shew that its area is to that 

 of the parallelogram as TT is to 2. 



3G. The least tangent to an ellipse intercepted by the axes 

 is divided at the point of contact into two parts, which 

 are equal to the semiaxes respectively. 



37. Find the area and position of the greatest triangle that 



can be placed in a given parabolic segment, having the 

 chord of the segment for its base. 



38. Find the least triangle which can be described about a 



given ellipse, having a side parallel to the major axis 

 and having the other sides equal. 



The height is three times the semi-minor axis. 



39. Prove that of all circular sectors described with the 



same pei-imeter, the sector of greatest area is that in 

 which the circular arc is double the radius. 



40. A chord PSp is drawn through the focus S of an ellipse, 



and the points P, p, are joined with the other focus H : 

 determine when the area PHp is a maximum. 



Let e be the eccentricity of the ellipse and 9 the 

 angle between the chord PSp and the major axis of 

 the ellipse. If 2e* is greater than 1 the maximum is 



determined by cos 2 = 2 ^ , and 0= - gives a mini- 



6 a 



mum; if 2e 2 is not greater than 1 the maximum is 



7T 



when 0*= , and there is no minimum. 



2 ' 



