J.~2 ANALYTIC GE<)Mi:iHY [Ca \. 



and not dependent upon the value of c. It ./' }>< <li\ i<l< <1 l>\ 

 //'. the c is eliminated from the equations (6), giving 



7 p- ... (7) 



Therefore the coordinates of the middle point <>t every 

 chord of slope m satisfy the equation 



or, y=--^x; . . . 



\\hich is therefore the equation of the diameter bisecting 

 tlu- chords of slope m. 



The form of equation [62] shows that every diameter of 

 the ellipse passes through the center. 



153. Conjugate diameters. Since every diameter passes 

 through the center of the ellipse, and since, by varying tin- 

 slope m of the given set of parallel chords, the corresponding 

 diameter may !< made to have any required slope, therefore 

 it follows that every chord which passes through the center of 

 an ellipse is a diameter, corresponding to some set of parallel 

 chords. In particular, that one of the set of chords given 

 by equation (2), Art. ir>:>, which passes through the cent* T, 

 i.e., the chord whose equation is 



is a diameter. This diameter bisects the chords parallel to 



the line [62] ; for if m 1 be the slope of the line [62], 



then ^= 



hence, mm 1 = - ] . . pJ4] 



