216 ANALYTIC QKOMRTnr [Cn. VIE, 



These deduction^ from .|u:ition [.".:.] show that the hyperbola has 

 the form represented in Fig. 1)8, and that, as increases from to o 

 lower half /I'lfof the infinite branch at the left is traced; as increases 

 from a to 360 a, the right hand branch VA U is traced; and as in- 

 creases from 360 -a to 360, the upper half .S.!' of the left hand Waneh 

 is traced. 



If increases beyond 300, the tracing point moves along the > 

 tl.is is also true if $ changes from to - 360. 



\"TK. To show the identity of the curve as traced iu the present 

 article and in Art. 117, it i, .! <>uly be recalled that 



Vo* + ft* ft* 



f - Z and that / = 

 a a 



These values substituted above show that 



itan- 1 ''Y that OA = - (a + W/' + ft), etc. 



EXERCISES 



1. From equation [55], trace the parabola. 



2. From equation [55], trace the ellipse. 



3. \\\ means of equation [55], prove that the length of a chord 

 through the focus of a parabola, and making an angle of :>" \\ith tho 

 axis of the curve, is four times the length of the latus-rectum. 



4. By transforming from rectangular to polar coordinates, derive the 

 polar equations of the conic sections from their rectangular equations. 



EXAMPLES ON CHAPTER VIII 



1. Find the equations of those tangents to the conic 9z* 16y*= 1 ! 1. 

 which pass through the point (0, 1). 



2. What is the polar of the point (7, 2) with reference to the e 



16 y* -f 9x* = 144? Find the equation of the line which is tangent to 

 the conic and parallel to this polar. 



3. Find the polars of the foci of the ellipse -f ^ = 1, with 

 regard to this ellipse. Also for the parabola y 9 = 4 px. 



4. What is the equation of ths polar of the center of the conio 

 A& -I- By* -f 2 Gx + 2 Fy + C = 0, with reference to the conic ? 



5. What is the pole of the directrix of the hyperbola **-4y 2 =16, 

 with reference to that curve? 



