CONTACT. CURVATUEE. EXAMPLES. 355 



28. Find the evolute of the curve p* = r* a 2 . 



29. If A be the area between a curve, its radius of curvature, 



and its evolute, then 



dx tfy 



2 dx* 



30. If p be the radius of curvature of a curve, then the radius 



of curvature of the evolute at the corresponding point 

 dp 



18 PJ- 

 r ds 



31. If x, y be the co-ordinates of the centre of curvature of 



the curve y* = c?x, shew that 





32. Shew that in a parabola the radius of curvature at any 



point is equal to twice the portion of the normal which 

 is intercepted between the point and the directrix. 



33. Investigate the following expressions for the radius of 



curvature at any point of an ellipse : 



(rr'}* b* 



(1) ' i > (2) . o ,,a> 



a (1 e 2 sm 2 <) 8 



where r and r' are the focal distances of the point and 

 </> is the angle which the normal at the point makes 

 with the major axis. 



34. The locus of the centres of all ellipses having the 



directions of their axes given, and having a contact of 

 the second order with a given curve at a given point, 

 is a rectangular hyperbola passing through that point. 



35. Find the asymptotes of the evolute of the curve 



y = a tan x. 



36. Shew that corresponding to the portion of the curve 



a s y 2 = x s near the origin, the evolute is approximately 

 a curve whose equation is xy* = c 3 . 



AA 2 



