1 \.\I.YT1C GEo.\li:i I;Y j ,,. X. 



and the point of intersect ion /' i* on tin- lino, / ,-, th, din-rii -\\ 



corresponding to the focu* i'. f Similarly, if tin- chord passes 

 <>cus F, = ( ae, 0), the point P' is on the directrix x = 



I/if tangents at the extremities of a focal chord intersect upon (he 

 carresjMtuliny ilirtctrit. 



Again, the Hue joining the intersection P* s f -, |M to the focus has 

 the slope 



<- 

 while the slope of the focal chord (1) is 



hence m' = , 



m 



and therefore the line joining the focus to the intersection of the tangents at 

 the ends of a focal chord is perpendicular to that chord. 



150. The locus of the foot of the perpendicular from a focus upon a 

 tangent to an ellipse. Lot the equation of a tangent to the ellipse 

 (Art. 11;'.), whose equation is 



7-2 ,2 



Si + P" 1 - ' ' < ] > 



be written in the form y = mx + Va 2 * + fr 2 . . . . (2) 



Then the equation of a perpendicular to (2), through the focus (ae, <> 



i 

 y = - (*-<), ', x + my = ae. . . . ( 



Tf 7 y = (x / , y') is the point of intersection of (2) and (3), it is re- 

 quired to find the locus of P ; i.e., to find an equation which will be j 

 satisfied by the coordinates x', y 7 , whatever the value of m; this i 

 be an equation involving x* and y, but free from m. Since 7* is on 

 both lines (2) and (3), 



therefore / - m^ = Va'm 8 + 6*, ... (4) 



and yf -f my* = ae. . . . (5) 



