146 AN ALT 'IK QXOMJCTBT [c... vn. 



Now let P, move along the circle until it coincides with 

 PI, t.e., until Xy=Xn and y^y^ ( ' lrn '<iuation ( (} ) 



which, by clearing of fractions and transposing, may be 

 written in the form 



.e., 



which is the required equation of the tangent to the circle 

 a? + y 2 = r 2 , ^ and yj being the coordinates of the point of 

 tangeiK \. 



() Center of circle not at origin. If the equation of the 

 given circle be 



a? + # 2 +2#* + 2 J fy + <7=0, . . . (7) 



then, P l and P 2 being on this circle, 



and * 2 2 + y 2 2 + 2#2: a + 21fy 2 + (7 = 0. ... (9) 



Subtracting equation (8) from equation (9), 



x* - x* + 2#(* 2 - *!> + y 2 - y x 2 + 2 P(y 2 - ^)= 0, 

 which may be written in the form 



whence, 



Substituting this result in equation (2) gives 



4- 



