DiriT.KKNTIAL CALCULUS. 20!) 



1001. A curve is generated by a point of a circle which rolls 

 along a fixed curve : j.n.ve that the diameter of the circle through 

 the generating point will envelope a curve similarly generated by 



a riivle of half tin- dimensions. 



1002. OY is a perpendicular let fall from a fixed point on 

 any one of a series of straight lines drawn according to some fixed 

 law ; prove that when Y is a maximum or minimum, Y is in 

 general a point on the envelope : and, if T be not on the envelope, 

 the line to which OY is the perpendicular is an asymptote to the 

 envelope. 



1003. On any radius vector of the curve 



r = a sec" 

 n 



u'-t.-r i- described a circle: the envelope of such circles is 

 tin- curve 



r = c sec"" 1 r . 



n-l 



Prove this property geometrically in the cases when n = 2, nml 

 \\liru n = 3. 



1004. I .n\<l>i>< of the system of circles represci 

 by the equation 



(x - am*)* + (y - 2am) f = o f (1 + m 1 )' 

 for different values 



1005. The envelope of the straight 



i 

 a: coe < + y sin < = a (cos n^>)" 



i ve whose polar equation is 



nB 

 r 1 ' = a 1 """COST . 



1-71 



1006. Tangents drawn to a series of confocal conies, nt points 

 < they meet a fixed straight line through one of the foci, 

 lope a parabola, -f traight line is dii 



n focus the focus. 



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