42 



ON THE EQUATION OF CUBVE8 



(/3.) When B^=AC the roots of equation (a) are equal 

 and the value of each is — B:A; hence, we see that the only 

 straight line which can be drawn from a given point so as to 

 cut a parabola only in one point is the diameter which passes 

 through the given point. 



(y.) When B'^<AC the roots of (a) are imaginary; and 

 therefore no straight line can be drawn in the plane of an 

 ellipse so as to cut it in only one point. 



VI. 



If the straight line (2) meet the curve (1) in the points Pi 

 and Pj = we have seen (i.) that 



_ F\^» + 2OTCOSy + l) 



Let Qi CI2 (Pig. 5) be a chord parallel to Pi P2, and passing 

 through the origin of co-ordinates O ; then, since F' becomes 

 F when A coincides with O, we shall have 



' ' AW^ + 2Bm+C ' 

 and by dividing the former equation by the latter we obtain 

 AP.AP, _F- 



oa,.oQ^ ■" F ^' 



Similarly, if P3P4 and Q3Q4 be another pair of parallel 

 chords, passing through A and O respectively, we shall have 



AP3^AP^_j;; 



'OQ3.OQ ~ F» 

 and by comparing this with the last equation we get 



AP,.AP, OQ,.QQ, 



AP3.AP^-OQ3.0Q^ ^^*'- 



Hence if two chords 0} a curve of the second degree be drawn 

 intersecting each other, the rectangle contained by the seg- 

 ments of the one will have an invariable ratio to the rectangle 

 contained by the segments of the other, provided that each of 

 the chords always remains parallel to a given straight line. 



