292 A. Miikhopadhyay — Memoir on Plane Analytic Geometry. [No. 3, 



§. 4. Coordinates of intersection of two lines. The following 



metliod of investigating the condition that the general equation of the 

 second degree 



ax''+2hxy-\'hij^ -^2gx-^2fy-\-c = 

 may represent two right lines, is shorter than the proofs usually given, 

 and has, besides, the advantage of furnishing at once the coordinates 

 of the point of intersection of the lines represented by the equation. 



Let (x', y') be the point of intersection of the lines ; removing 

 our origin to this point, the equation becomes 



ax^ -\-2hxy-\-hy^ -{■2g'x-{-2fy-\-c =:0 (1) 



where g' = ax' -f hy' + g^ 



f =^h:,'J^hy'-\-f, 



c' = ax'^ ■^2hx'y' -\-hy'^ ■^'2gx' -^2/1/' -^-c. 

 But the equation (1) now represents a pair of lines through the origin, 

 and, as such, it ought to be homogeneous in the second degree ; therefore, 

 each of the quantities g', /', c' must vanish separately, which gives 



aar'-^hy'-[-g = (2) 



7.^'+V+/ = (3) 



ax'\2hx'y'-\-hy'''-\-2gx'-Ji-2fy'-\-c = (4) 



Multiplying (2) by x', (3) by y', and subtracting the sum of the pro- 

 ducts from (4), we get 



gx'-\-fy'-^c = (5) 



From (2) and (3), we have 



^-^6^r^' 2/=^jTr3' (^) 



which are, accordingly, the coordinates of the point of intersection 

 of the lines represented by the given equation. Eliminating a;', y', 

 from (2), (3), (5), we have the condition that the discriminant must 

 vanish in order that, the equation may represent two right lines, viz.^ 

 a h g 



h h 



9 f 



= (7) 



ax^'{-2Jixy-\-hy^-\-2gai-\-2fy-\-c = 



As the equation 



is transformed to 



ax^-^2J2xy-{-hy^=0 

 when the axes are removed to the point of intersection of the lines, it 

 follows that, as the angle between the lines is not altered in magnitude 

 by the transformation, the angle between the lines given by the general 



