H-11V.J //// < ON// s/.i //- I'.,; 



7. with crnUT at thf i . Mini-Aiea and 2, and the 



ansvenw axis parallel to the y*xi - cunre, 



a Find the fuci and latus rectum for the hyperbola* of 



9. Hy a uitable transformation of coordinates, reduce the equation* 

 of exercises 6 and 7 to the standard form - - = 1. 

 I the foci of the hyperbolas 



(u) a-?" 1 - w r-f- 1 - (y) ?-f' L 



Plot UM ennret 0?) and . 



119. Every equation of the form Aa& + By* + t Ga + 1 /V 

 < 0, in which .1 and n have unlike signs, represents aa 



hyperbola whose axes are parallel to the coordinate axes. 

 u cleared of fractions and expanded, tin- three equations 



found t.>r the hyperbola are of the form 



A#+Bf + *Qx + *F9+C=". . . (1) 

 where A and B have opposite i</n, and neither of them it zero. 

 versely, it will now be shown that every equ this 



i represent* an hyperbola, whose axes are parallel t- > 



ae axes. A numerical case will be examined first, 

 anil then the general equation. 



VMPLE. To dhow that the equation 0jr*-4j a -18jr + 24y-63=0 

 reprtMcnU an hyperbola, and to find ita elements. Tranaposiiig the con- 

 stant term, and completing the squares of the x-terma and jr-terms, the 

 equation may be written 



oe this equation is of the form [48], its locus hat the geometric 

 operty given in Art. 118, and therefore represents an hyperbola. IU 

 nter is at the point , 1. .) its transverse axis is parallel to the x-axis, 

 length; 4, and its conjugate axis is of length 6. The eccentricity It 

 = i Vl3, the foci are at the points (1 - via. 3) and (1 + Via, 3); and 

 e directrices are the lines whose equations are 



