iss .1 \. i /.)//( QMOMXTBT \ m. 



that is, 



AF*-ABC 



V AB 



which heroines, if the second memher ! n-jin-srni. A. 



. < 





Comparing tins r.pmtion with [4~>] or [46], it is seen tx 

 rxjuvss tlu geometric relation of Art. 11-. and therefor 

 r presents an ellipse. Its axes are parallel to the co"idinat< 



axes, its center is at the point ( -, -^A and the length 



\ A BJ 



of the semi-axes ; 



The foci and directrices may be found as above. 



NOTE. If A = B, then equation (1) represents a circle (Art . 79). If 

 ABC>BG* + AF 2 , equation (1) having been written with .1 an<l H 

 positive, then no real values of x and y can satisfy equation (_' >. u hi<-h 

 is only another form of equation (1), and it is said to represei 

 imaginary eltipxe. If ABC = BG* -f AF*, then x = - Q, and y = 



A. ' * 



are the only real values that satisfy equation (2) ; in that case, this equa- 

 t KMI is said to represent a point ellipse; or, from another point of 

 two imaginary lines which intersect in the real point f - , - j. Each 

 of the above may be regarded as a limiting form of the ellipse. 



EXERCISES 



Determine, for each of the following ellipses, the center, semi-axe 

 foci, vertices, and latus rectum ; then sketch each curve. 





