OF THE SECOND DEGREE. 



hence, after slight reductions, we get, 



for the ellipse, -^ + ^ = 1 W, 



49 



for the hyperbola, — ^ = 1 {SO). 



o ■* ^ 



These, therefore, are the equations to the ellipse and hyper- 

 bola, referred to any system of conjugate diameters as axes of 

 co-ordinates. 



O.) Taking now A as the origin of co-ordinates, let AQ=x, 

 Q,F—y, be the co-ordinates of P; then, for the ellipse, 

 equation (28) gives 



f : X {%a'—x) : : h"" : a"", or 



,f= ^^{2a'x-x^) (31); 



a 



b«t, when the curve is a hyperbola, we have 



y^ : X {2a'+x) ::b'^: a'\ or 



y'=^,{a2'x+x^) (32). 



Equations (31) and (32) are the equations to an ellipse and 

 hyperbola, referred to any diameter and a line drawn through 

 its extremity parallel to its ordinates as axes of co-ordinates. 

 The values of the constants a and b' which occur in the last 

 four equations are given by equations (c) and (d) of No. viri. 



Returning to equation (e) of No. i., it is evident that when 

 the co-ordinates Xi y^ of the point A satisfy; the conditions 



F=o, D'm+E'=o ....(a), 



each of the roots of equation (e) will be zero, and conse- 

 quently the straight line (2) will be a tangent to the curve (1). 

 The former condition, F'=:o, merely implies that the point 

 A should be on the curve (1). From the latter we obtain 

 «»= — E' : D', which substituted in equation (2) gives 



^'{y-Vi )+E' ix-x, )=o (b) 



for the equation of the tangent applied to the curve (1) at the 

 point xi yx , From equation (b) we readily deduce 



u 



