356 NOTE ON THE INTERSECTION OF TWO CURVES. 



intersection and the points for which ro = — )\,0.,=^i8o° -\-^j^\iQ 



"t)n = e cos -\- e' sin j?, another chord through two points 



r 

 •of intersection. Both these chords pass through the intersection 

 •of the directrices corresponding to the common focus. 



2. The points of intersection of /(>, 0)=^ o, F{r, ^)=omay 

 be completely given by finding the points on the two curves for 

 which the ;- and the ^ are the same. 



For example, r = 2 a cos 0, r cos () = a have all their inter- 

 sections given in this way. In their cases, the points {)\, ^■^), 

 (-fi, i8o° + c/i) are both given on the curves, and each equa- 

 tion is unaltered by writing — r for r and i8o° + $ for (j. 



I 

 Again, tlic curves — r= i — c cos fi. >' = -o. cos (y have all 



r 



their intersections given in this way. These points are given by 



2ae cos -0 4" 20 cos = 1, and the roots of this equation in ^ are 



a, 360° — a, /], 360° — fS, say ; for each there is a corresponding 



value of r given by ;- = 2a cos 0. The points for which r, = 



/ 

 — r^, 0., = 180° + 0^ are given by — = i 4- c cos 0^^. r.. = 2a 



ri 

 cos ^\, I'o = —i\, 0. — 180° -f 0^, 



I, 

 ..-. — = I -\- e cos 01, — r^ = 20 cos (180° + 0^), i.e., 



)\ i\ = 20 cos 0-^, 



. • . the equation for 0.^ is the former equation for and the points 

 of intersection found are the former points. 



In general, if the substitution of — r for r and 180*^ -f B for 

 ^ leaves the equation of one of the curves unaltered, say F {r, 0) 

 = o. all the intersections are given in this way. For the other 

 points would be given by — 

 / (''1, 0i) =0, F (r„ O.) =- o, r. = — i\, 0, = 180° -f 0„ 

 • •• by / (r„ 0^) =o,F {—i\, 180° + 0,) = o, 

 ••• by / {}\, 0^) = o, F (ri, 0^) = o, 

 which gives the points already given by f (r-^, 0^) =z o, F (r,. ^o) 



— o, r^= rj, 0., = 0^. 



3. Return to the example given in Section i. Two points of 

 intersection are given by el' cos ^ — c'l sin =^ I — /, '(A), and 

 therefore are real points if 



e-r- + e'-i- > (i — ry, 



a condition which must be satisfied if the conies are parabolas or 

 hyperbolas. The other two are given by cl cos — c'l sin ■= 



— / — I', (B), and therefore are real points if 



c^l'-^ j^ e'-^l^ > (/ + l')\ 

 a condition which cannot be satisfied if the conies are ellipses. 



A distinction can be drawn between these pairs of points 

 in the case of the intersection of two hyperbolas. If the first 

 conic is traced by taking values of from 0° to .360°. for all 



