352 CONTACT. CURVATURE. EXAMPLES. 



2. In the curve 



y = x 4 - 4# B - 18# 2 , 



the radius of curvature at the origin = -fa. 



3. In the curve 



y = x 3 + 5a? + Go;, 



the radius of curvature at the origin = 22.506... 

 Find at what point the radius of curvature is infinite. 



4. If < (x, y} be the equation to a curve, then 



dy 



"T- i ~ T_ __ n - T - T - T I [ r i r 



rf^/ da? dx dy dx dy \dx) dy* 



5. Find the parabola whose axis is parallel to that of ?/ 

 which has the closest possible contact with the curve 



x 3 



y = - - at the point where x a. 

 d~ 



T> i. f ^A 2 / a\ 

 Result. (*- 5 J-3(ir-*i). 



G. If r = a(l cos^), p = sin-. 



o z 



IT r* fn a t\ a (5 4 cos 6y 



. , , 



8. If the curves f(x, ?y) = and < (x, y) = touch, shew that 

 at the point of contact 



_ 



dx dy dy dx 

 9. Apply the last result to find if the straight line 



a 

 touches the curve 



10. When the angle between the radius vector and the per- 

 pendicular on the tangent has a maximum or minimum 

 value, shew that pp = r*. 



