534 DISCRIMINATION OF THE CONIC. 



S =--= ax^ -f- 2 hxy -f hy^ -f 2^^/.r -f- 2/v + r = o ( i ) 

 is then taken in hand. Its slope is 



nv -4- hv 4- n 



(2) 



dx 

 where X is an}' constant, is a straight line joining points on the 

 curve, at which the slope is the same. For all vakies of X the 

 line passes through the intersection of the lines 



hS hS 



— - = o and -^ —o. (4) 



ox oy 



It follows from considerations of symmetry that the inter- 

 section of these lines must be the centre of the conic. Solving 

 etjuations (4). the co-ordinates of the centre are obtained 



- 16/1 - f ^^ 



I a h 

 '// h 



ah 

 hi) 



(5) 



Non-central Conic. — If ab = h'-, then the centre of the conic 

 recedes to infinity ; that is the case of single symmetry. Equa- 

 tion (I) takes the form 



(ax + ^y ) \ H- 2gx -f 2fy + r = o, (6) 



while its slope at any point is 



d^- V ^ ' ..R ,--U R-^,.-i.-fl (7) 



Parallel Straight Lines.— When g and / both vanish, the 

 slope of the non-central conic is single-valued and constant ; 

 hence it degenerates into a pair of parallel straight lines. 



Parabola: Its Vertex and Axis. — When g and/ do not 

 vanish simultaneously, the slope is a function of the co-ordinates, 



. , . ci 



and the curve becomes a parabola. The slope at mhnity is ^ ; 



the axis of the curve is therefore parallel to the line /3 v + o.r 

 = o. Chords perpendicular to the axis are given by ay — (Sx 

 = yu. The value of ft. which makes this line a tangent, is 

 •easily found to be 



{f^^gay--c(a^-\- ^^Y 

 2{fa~gl3) (a-^-f ^'). 



This gives the tangent at the vertex, and from it the co-ordinates 

 of the vertex, and the equation to the axis are easily obtained. 



