AftAl.YII' <.i:<>.Ml-:i l;Y VIII. 



of theellip- . divide l*% internally and externally in the ratio f ; hen. 

 80) these coordinates are (V. ~i) ( <-\ the center 



of the ellipse, is the point (|, ~V) l! ''"' ""u r m IK; next transfoi -in ! 

 to tin- poim c. ill.- equation will be reduced to the first standard toim. 



in 



Since the axis A A is of length , and the eccentricity is jj, tin 



. 

 ax*s an- - and 2; hence the reduced equation, with C as origin and 



V5 

 CA as j-axis, will be 



The problem of reducing to standard form the equation of an ellipse, 

 when the directrix is not known, will be postponed to Chapter XII. 



EXERCISES 



Find, and reduce to the lir>t standard form, the equation of the ellipse: 



1. with focus at the point (1, ~3), with the line x + y = 7 for direc- 

 trix, and eccentricity J ; 



2. with focus at the point (a, //), the line - -f ^ = 1 for directrix, 



j a 6 



and eccentricity (where /<n). 



H 



III. THK HVI'KRBOLA 



Special Equation of the Second Degree 

 Ax* By* + 2Gx + ZFy + C = 



115. The hyperbola defined. An h\ jxTlx.la is tli<> Inn; 

 a point which moves so that the ratio of its distance from a 

 fixed point, called the focus, to its distance from a li\ <1 line, 

 called the directrix, is constant and greater than unity. T In- 

 constant ratio is the eccentricity of the hyperbola. This 

 curve is the conic section with eccentricity e > 1 (cf. 

 Art. 48). 



