J10 BOOK OF MATHEMATICAL PROBLEMS. 



1007. If a parabola roll along a straight line, the envelope of 

 its directrix is a catenary. 



1008. A parabola is described touching a given circle, and 

 having its focus at a given point on the circle : prove that the 

 envelope of its directrix is a cardioid. 



1009. A straight line is drawn through each point of the 

 curve r m = a m cos mO at right angles to the radius vector : prove 

 that the envelope of such lines is the curve 



r n = a n cosnO J [n (1 - m) = m}. 



1010. ST is the perpendicular from the pole on a tangent to 

 the curve r m = a m cos mO ; with S as pole and Y as vertex is de- 

 scribed a curve similar to r n = a n cos nO : prove that the envelope 

 of such curves is the curve r p = a p cospO, where 



Ui+I+1. 



p m n 



1011. If A be the vertex, P any point of the parabola 

 y* 4ax t the straight line through P at right angles to AP will 

 envelope the curve 27 ay* = (x 



1012. A circle is described on each radius vector of a given 

 curve : prove that the envelope is the locus of the foot of the 

 perpendicular from the pole on the tangent. 



