1887.] A. Mukhopadhyay — Memoir on Plane Analytic Geometry, 295 



pai 



r with the line 

























\x-^fiy-\ri 



' = 



IS 



















a 





h 



\ 



a 







h 





h 



M 









9 





f 



V 







h 





I 



1 



h 



h 



/^ 







a 





h 





a 

 \ 



h 



X 









(21) 



The length of the portion of Xaj+/A2/+v = which is intercepted 

 between the pair of lines is also easily found ; for, from (12), the per- 

 pendicular from the point of intersection of the pair of lines on 



Xx-\-iJLy-^v=zO 

 is at once seen to be 



a h \ 



h h IX 



g f V 



|(fe«-a&)(X« 



Hence, the length of the intercepted portion is 



(22) 





1 



a h 



X 





X -M* 



h b fx 





„ /^ ^ 



9 f ^ 



(9^\ 



2 h h 



1 



2 



h b fx 



\^o) 



a h 





a h \ 







\ fx 





The product of the two sides is, by a glance at (20), written down 



to be 



1. 



a h X 



« 



2h a-h ' 



h b fX 





h-a 2h 



9 f 



/ r% t \ 



h a " 



h b fx 



- ... (24) 



b h 



a 7i \ 







X 



IX 







As an application of the formula in (21), we can find the area of 

 the parallelogram formed by the two lines 



ax^-\-2hxy-\-by^-^2ga:-{-2fy-^c=0 

 with ax^-\-2hxy-\-by'^ =0 



which are two lines through the origin parallel to the first pair. By 

 subtracting the equations, we see that 



2gx-\-2fy + c=:0 



