154 



\\.ll.YTK' '.I 'M//,-/7.T 



' M. VII. 



90. Chord of contact. If two tangents are drawn from 

 external point to a circle, the line joining the two c 

 spending points of tangency is called the chord of contact fr 

 the point from which the tangents are drawn. 



The equation of this chord of contact maybe found l>y 



first finding the points of tan- 

 gency and then writing tin- 

 equation of the straight line 

 through t li< isc two points. It 

 may, however, be found 



briefly, and much more ele- 

 gantly, as follows: 



Let P 1 = (x v y^) be tin- 

 given external point from 

 which the two tangents are 



drawn; and let T^ = (x v y a ) and T z = (x v y 8 ) be the points 



of tangency on the circle 



2^ _|_ ^2 _j_ 2 G x 4. 2 Fy -f (7 = ; . . . ( 1 ) 



it is required to find the equation of the line passing through 



r a and T z . The equation of the tangent at T^ is (Art. h 1 ) 



and the equation of the tangent at T 3 is 



x&+y&+G(x + x*) + F(y+y^ + C=Q.. . .(3) 



But each of these tangents passes through the point 

 hence ils coordinates, x and y v satisfy equations (2) and 

 then ; 



^yijfi-n ^(i + *i) + ^(yi + y,) + ^=o,.. . (4) 



and x& + y^ 8 + 0^ 4- * 8 ) -h^! 4- y 8 ) + C = 0. . . . 

 Equations (4) and (5), 1 1 owe ver, assert respectively 

 (x v y^) and (r 3 , y 8 ) are points on the locus of the equation 



=0. . . . (ti) 



