ANALYIK <,/ :<>MI:I i;y [c.. X. 



the two foci coincident at the center, an<l tin .lii-.-.-iri.-:s 

 in finitely distant. 



' V 



The equation -V + *rr = l represents the polar of the 



point (* p y } ) with respect to the ellipse; if the j...int is 



outside the curve, this polar line is its < h<>nl of contact; 



if upon the curve, the polar is the tangent at that point 



i te. 122, 126, 127). 



These facts will be assumed in the following work. 



143. The equation of the tangent to the ellipse ^5 + ^=1 



& It" 



in terms of its slope. The equation of a line having t In- 

 given slope m is y = mz + k; . (1) 



it is desired to find that value of k for which this line will 

 become tangent to the ellipse whose equation is 



5+S- 1 - 



Considering equations (1) and (2) as simultaneous, and 

 eliminating y, the resulting equation 



(fc 2 + a 2 2 )* 2 + 2 cfonkx + a 2 #> - a 2 // 2 = . . (3) 



determines the abscissas of the two points of intersection of 

 the curves (1) and (2). When the curves are tangent, these 

 abscissas are equal ; therefore 



i.e., 

 and 



Hence y = mx^/a?m* + l>' 2 . . . [.V."| 



the equati 

 values of m. 



is the equation of a tangent to the ellipse -- + ^ = 1, for all 



a* cr 



