88-80. J Tilt: rntCLK \i , 



> ; these \ .linen of m, being sob- 



stitutrd in turn in i-qnation (-), give tli. uugcfiU 



/', to ti;. ( 1). 



lienar^ + y^-fJ.O; hence the 



>ies of m from equation <j .,-, a n<l the two 



UngenU also coincide, i.e., there is in thin cane but on* 



tangent. It l\ . within i .<, then > values 



liom eqnat .ire l<>th imaginary ami no tangent 



rilWU U) tin- ')* 



1) is Mihsiitiitcd in 



equation < _' , and tin n equations ( -j > an<l ( 1) are considered 

 as ni inn 1 lain r and y, the coordinate** of 



the corresponding point of contact are obtained. 



> re. The properties of the e?uafi<wu of the line and circle hare thtu 

 established a geometric property of the circle [cf. Art. 31, (III)]. 



If the equation of tin- tfiven circle liad been 



j* + y*+ liGr-f SJV-hC-O, ... (5) 

 ul.l. l.y Art. 71. IKIVC IMTH transformed to new axes 

 its center ( G, ~F) and parallel respectively to 

 the given axes; its equation would thus have become 



j' and y refer to the new axes. 



nuition, however, leaves the circle and all its 

 intrinsic properties unchanged ; but (a) applies to circle (6), 

 :.\vd thatcircli- it h is circle (6) merely 



tlu-r axes, has the same properties. 



Thou* roncliuiona may also be stated thus: If P\ to o*t*idf of the 

 circle, equation (4) gives two reU and distinct Taloes for m ; corrrspooding 

 to these there are two real and distinct tangenu ; if /S to on the circle, the 

 two ralnes of m are real but coincident, and there arc two real bat coincMeai 

 tangents ; if P l is fiirtfe of the circle, the two Taloes of m are 

 and the two onffi^t^Ki^g tangents are therefore also imaginary. 



