CONTACT. CURVATURE. EXAMPLES. 353 



dx b* 



11. If at every point of a curve 2a -j- = -- h y> then 



as y 



p = -=-*TI . Shew also that - + - = -, where n is the 

 y b n p a 



portion of the normal intercepted by the axis of x. 



12. Find the value of p when r = a cos 9. 



13. If#=Vc 2 + s 2 



14. The equations which determine the co-ordinates a, b, of 



the centre of curvature of a curve may be put in the 

 following form, where r 2 = a? + y* : 



, cPx dV a ,cPy eZV 



S) n . __ _ OA _ ^_ _ 



^ df 9 as? as?* 



15. In the parabola y* = 4mx, 



, 2# ? 2 (TO + a)t 



b= -- r- , p= v . - . 



Shew that the circle of curvature at any point of a 

 parabola, except the vertex, cuts the axis at two points 

 on opposite sides of the vertex. 



16. If Aa? + By*+C=0, 



AA-B , BB-A 



then a = 



17. If, 



18. The radius of curvature of the curve ?/ 2 = - - -. 



c-4a 



at one of the points where y = is , and at the 



ru 3a 



other . 



m 



19. If s = a sin" ^, find p. See Art. 324. 



20. Find the equation to the circle of curvature of the curve 



y* = 4aV x*, at the origin. 



T.D.C. A A 



