CONIC SECTIONS, ANALYTICAL. 1 ->7 



that <V. - % C0 ! ">' is equal to <-' tjP" j if 2a) 2/Jbe the axes. 

 (ab - c *) am 8 <o a*/2 a 



have then the eccentricity e given by the equation 



e* ( + 6-2c cos fa>)' 

 f ~ 



The foci may be determined from the condition that the rect- 

 angle under the perpendiculars from them on any tangent is 

 constant. Thus taking the simple case when the origin is the 

 centre, and the axes rectangular, if the equation of the conic be 



and (JT, Y) be one focus, (- A", - Y) the other, we must have, 

 in order that the straight line lx + my=l may be a tangent, 



- j - = a constant = /*, 



or, r(jjL + X*) + w f (fi + Y*) + 2lmXY- 1 = ...... (A). 



But the condition that this line may be a tangent gives also 

 tin- condition that 



ax 9 + by* + 2cxy -f(lx -f my) 9 

 must be a perfect square, or 



whence F1>f+ m'af- 2lmc'f+ c' f - db = ......... (] 



Now (A) (B) expressing the same condition must give tin- 

 same relation between I and m; hence 



/i4.r* zr i 



/ 



Tlic < are then 



