

ANA 1. Y IK GJSOM1 TRY 



VTIl 



thon fintl these coordinates by solving two rni:iti.! . >n<ling to 



equations (4) and (5) above. 



The problem of finding the above tangent could also have been solv <l 

 by writing the equation of a line through the point (~4, -1) (Art 

 and having the undetermined slope m, and then BO determining m that 

 tin- two points in \\hidi this line meets the parabola Him nl<l ! roinci 



12S Through a given external point two tangents to a conic 

 can be drawn. Tins thcuivm can !> prv<l in pn-< -isdy tin- 

 same way as the corresponding theorem in the case of 

 the circle (Art. 89) was proved. It may also ! proved b\ 

 the method already applied to the parabola in tin- preceding 

 article. Let the latter method be adopted. Suppose the 

 equation of the conic to be 



1 ; + pi + 2 Gx + 2F+ C=0; . . . 



let the locus of this equation be represented l>y the curve 

 /, and let <>=(A, k) l>e the given external point. 



If P 1 s3(a?,,y,) is a point 



Y /on LPiP 2 L r , then the equa- 



tion of the tangent at PI is 



X_ -r^wry U + C=QI (2) 



and this tangent will pass 

 through the point Q if 



But PI being on the locus of equation (1), its coordinates 

 and y l also satisfy equation (1); 



.e., 



If now equations (3) and (4) are solved for x\ and y v two 

 values of each are found; these values are both imaginary 

 if Q is within the conic, they are real but coincident ii 



