ll-tt.] 





141 



ugh the point of tangency in called a normal line to the 

 .1 ji.iiiit. Tim*, in Kitf. '!. /',/Y /',/*! are e- 

 TiA* tangent, and 1\N a normal to the curve at />,. 



. 



82. Tangents: Illustrative 



) To find the equ.t that tangent to the circle x*+j6 



h makea an angle of 45 with the r-axis. Since thb line make* an 

 angle of 45 with th. r-axia iU equation is y = x + *, where 6 U to be 

 determined ao that thi* line shall touck the circle. 



.irly, from the 0gure, there are two value* of b (OB l and OBJ for 

 . tin* line will be tangent to the 

 -. According to Art. 81. these 

 values of b are thoae which make the two 

 : ii.t.Tx-.tioi. of the line and the 

 le become coincident. 

 Considering the equations x* + y 1 = ft 

 ami y = x -- b simultaneous, and elimi- 

 nating y. tin- resulting equation in x U 

 (x+*)=5, i>, 2x-.2&r+6-5=0. 

 root* of this equation will become 

 abaciMM of the points of 

 nwction will become equal (Art. 9), 

 If ** - 2 (P - 5) & < 





I:., o. 



The equations of the two required tangent lines are, therefore, 

 y = x -- Vio, and y = x - VU). 



f t hose tangents to the circle r* + y 1 = 6 y 

 that are paralM to t i..- lim T + 2y + 11 = 0. 



The equation of a line parallel to r + 2y + 11 =OUx + 2y + lr = ( 

 re k is an arbitrary constant (Art. 02), and thU line will become 

 . the cin he constant i- be so chosen thai the 



li th.- line meets the circle shall become 001 



(4 the equations x 1 + y* = 6y and x42y-flr = simulta- 

 is, and eliminating x, the resulting equation in y is 

 ( - * - 2y) + f = 6y, ,>., 5y 4 (4 k - )y + lr = 0. 

 'he two values of y will become equal if (Art. 0) 

 - ) 

 the two required tangent lines are: 

 x + 'Jy- + 3>/o=O f and x -f 2y - 6 - 3v^5 = 0. 



