366 ENVELOPS. EXAMPLES. 



X* if 

 16. From every point of the ellipse j- 2 + ^ 1 = pairs of 



a; 2 if 

 tangents are drawn to the ellipse -^ + "^ 1 = : 



G/ i/ 



shew that the locus of the ultimate intersections of 



V, A f f ' , 1 



the chords of contact is j- + -fj- = 1. 



a b 



17. Circles are drawn passing through the origin having 



their centres on the curve a*y*b 2 ('2ax x z ) = Q: shew 

 that the locus of the ultimate intersections of these 

 circles is (a;* + y 2 - 2ax) z - 4aV - 4&y = 0. 



18. The circle whose equation is y? -f y 2 + 2ax + 2by + 2c = 0, 



is cut by another circle which passes through the 



x 2 if 



origin and whose centre is on the curve -5 + ^ 1 ' 



a. p 



shew that the chord joining the points of intersection 

 touches the curve aV + @ 3 y* = (ax + by + cf. 



19. Find the locus of the ultimate intersections of the 



straight lines 



y cos 6 x sin 6 = c c sin 6 log tan C- + - j , 

 where 6 is the variable parameter. 



X X 



Result, 2y = c(e* + e~c). 



20. The equation to a spiral is r n cos nd = a n ; straight lines 



are drawn through the extremities of the radii vectores 

 at right angles to them : shew that the envelop of these 

 straight lines is the curve 



ft 



r m cos m6 = a m , where m = - -. 



n + 1 



21. A series of ellipses has the same centre and directrix : 



shew that the envelop is a pair of parabolas, but that 

 the envelop will not meet those ellipses whose excen- 



tricity is less than . 



