56 



ON THE EQUATION OF CURVES 



or (A^i H-Bx, +D)2/+ (B^i +Ca:, +E)ar 



+ Dy,+Eari + F=o (33), 



which is a simpler form of the equation to the tangent applied 

 to the curve (1) at the point xi y^. 



(a.) Tlie equation DVi-J-E'=o may also be considered as 

 the condition that the straight line (b) may pass through the 

 point Xi y\; and therefore it determines the direction-index 

 m of the ordinates to the diameter of the curve (1) which 

 passes through a given point A. Hence we see that the 

 tangent applied to any curve of the second degree at a point A 

 is parallel to the ordinates of the diameter which passes 

 through that point. 



(^.) Let two straight lines be drawn from a fixed point Xi yy 

 touching the curve (1) at the points x'y' and x"y" respectively; 

 then, since each of these tangents passes through the point 

 ^1 2/ 1 } we shall have, by equation (33), 



{Ay' + Bx + D)2/i + W + Cx +E)«, + Dy' + Ear' + F=o, 

 {Ay" + Bx" + D) 2/, + (By" + Cx" + E)Xi + Dy" + Ex" + F=o. 

 But these are also the conditions that the points x' y' and x" y" 

 may be on the straight line 



(Ayi + Bxi + D)2/ + (Bt/i+Cari + E)jr+D^, + Ej:,+F=o...(34); 

 hence it is evident that the straight line (34) is the chord of 

 contact of two tangents drawn to the curve (1) from the given 

 point X\ y\. 



(y.) \i X yhe the point of intersection of any two tangents 

 to the curve (1), and Xiyi any fixed point in the chord of 

 contact, we shall have, by equation (34), 

 {Ay + Bx + D)3/, + {By + Cx + E)xi + Dj/ + Ex + F=o, 

 which can also be written in the form 



(A^,+Bxi4-D)y+(B3^, + Cx, + E)x+D5^,+Ex, + F=o...(35). 

 Hence we see that, if any chord of the curve (1) be drawn 

 through the fixed point Xi yx, and tangents he applied to the 

 curve at its extremities, the locus of the intersection of the 

 tangents is the straight line (35). 



