2f>i'. AMALTTK) GBOMfTBI [Cn. x. 



(5) Referring again to Fig. Ill, where ON is perpen- 

 dicular to the tangent at P r the conjugate diameters P t P a 

 and PJ PJ intersect at an angle ^ such that 



= 90 +Z / 



Hut, by Art. t',1, since the equation of the tangent at 



ab 





'i 2 +W JV,*V 



*T* 5*~ 



but CP l = ' . 



hence sin ^ = , . . (4) 



and the angle between two conjugate diameters is shr 1 -^. 



a o 



(e) Tangents at the extremities of a pair of conjugate 

 diameters form a parallelogram circumscribed about tin- 

 ellipse; its sides are parallel to, and equal in length to, 

 the conjugate diameters. Since the area of a paralh -lo^r am 

 is equal to the product of its adjacent sides and the sine of 

 the included angle, therefore the area of this circumscribed 

 parallelogram is 4a'b f simjr, which, by (4), equals 4 ab. 



That is, the area of the parallelogram constructed upon <///// 

 two conjugate diameters is constant; it is equal to the area of 

 the rectangle upon the axes. 



() A simple relation exists between the eccentric an- 

 of the extremities of two conjugate diaim-t 



Let the eccentric angle of P l = (x r y^ he fa (Fig. 112), 

 and of PI = (x y y,) be fa ; then the slopes of the conjugate 

 diameters may be written (cf. Art. 146), 



