MISCELLANEOUS EXAMPLES. 



o. Trace the curve x'l/ 1 + (x* a?) (x* b 2 } = 0, and shew 

 that the breadth of each closed portion is twice as great 

 in the direction of y as in that of x. Shew also that 

 when 5 approaches a as its limit, each of these portions 

 is ultimately similar to an ellipse. 



6. Trace the curve (x* - a 2 ) 2 + (t/ 2 - V'f = a 4 . Shew that 



when b = a it reduces to two ellipses. 



7. If a conic section whose focus is at the pole of a given 



curve have with the curve a contact of the second 

 order at the point (u, 6} the equation to the conic sec- 

 tion will be 



du ] 



* 



ff 7r<' 



cos (6 6)j dti 



8. A given curve rolls on a straight line, explain the 

 method of finding the locus of the centre of curva- 

 ture at the point of contact of the curve and straight 

 line. 



If the rolling curve be an equiangular spiral the re- 

 quired locus will be a straight line ; if a cycloid a 

 circle ; and if a catenary a parabola. 



.'). Right-angled triangles are inscribed in a circle : if one 

 of the sides containing the right angle pass through* 

 a fixed point, find the curve to which the other is 

 always a tangent. 



Result, c 2 (z? + y 1 } = (a 2 + 6 2 - c* - ax - %) 2 , 



where a and b are the co-ordinates of the centre of the 

 given circle and c its radius, the fixed point being the 

 origin, 



10. Determine the equation to the envelop of all the equi- 

 ' lateral hyperbolas which have a common centre and 

 cut at right angles the same straight line. 



Result, x* + 3 (axy}* - if + a* = 0, 

 where x = a represents the given straight line. 

 T. D. C, E E 



