ANALYIK >,i:o.\i /;//;>' [Cn. \ ill. 



\N Inch is the equation of the hyperbola, with a and b as snni- 



axrs. 



Again, if the transverse axis is parallel to the y-.\\\^ as in 

 Fig. 89, with tlu' renter at tin- i'>int (A, A;), the iMjuatiun <>1 

 tin- hy].rrl>olii will be found to be 



ii 1. That the expressions obtained on p. 103 for the distances 

 from the center to the focus and the directrix, of hyperbola [47], are 

 equally true for hyperbolas [48] and [49] follows from the fact that, 

 those expressions involve only a, ft, and c; moreover, equation (4) of 

 Art. 116 determines e in terms of a and ft; hence, for all these h 

 bolas, c a = + , the distances from the center to the foci are given by 



a- 



ae = 

 and those to the directrices by 



NOTE 2. It should be noticed that in equations [47], [48], [49], tin- 

 negative term involves that one of the coordinates which is parallel to 

 the conjugate axis. 



EXERCISES 



1. Find the equation of the hyperbola having its focus at the point 

 (-1, ~1), for its directrix the line 3ar y = 7, and eccentricity f. Plot 

 the curve (cf. Art 102, and Art. 109, Ex.). 



Find the equation of the hyperbola whose center is at the origin and 



2. whose semi-axes equal, respectively, 5 and 3 (cf. Art. 116, [47] ) ; 



3. with transverse axis 8, the point (_'n. .") heing on the curve; 



4. the distance between the foci 5, and eccentricity \ _' 



5. with the distance between the foci equal to twice the transverse 

 axis. 



Find the equation of an hyperbola 



6. with center at the point (3, ~2), semi-axes 4 and 3, and the trans- 

 verse axis parallel to the z-axis. Plot the curve (cf. Art. 118) ; 



