364 ENVELOPS. EXAMPLES. 



EXAMPLES. 



/j* nt 



1. Find the envelop of the series of straight lines - -f 1 v = 1, 



where *Ja + *Jb = */k a constant. 



Result, x^ -f y^ $. 



2. Ellipses are described with coincident centre and axes, 



and having the sum of the semiaxes = c. Shew that 

 the equation to the locus of ultimate intersections is 



fi a 

 + # C 5 . 



3. Find the envelop of all ellipses having a constant area, 



the axes being coincident. 



Result. 4sx*y* = c 4 where TTC* is the given area. 



4. A straight line cuts off from the co-ordinate axes distances 



AB, AC, such that nAB + A = c, shew that the 

 envelop of the straight lines is 



(y 4- nx c) 2 = &nxy. 



5. Find the evolute of a parabola y z = ax, by the method of 



Art. 337, taking the equation to the normal in the form 



y = m (x 2a) am 3 . 



Result. 27ay z = 4 (x - 2a) 8 . 



6. Find the evolute of the curve x* + y% = cfi. See 



Example 9, to Chapter xvni. 



Result, (x + 7/) ? + (x - y}% = 2a ? . 



7. Shew that the envelop of the series of parabolas 



under the condition db = c 2 , is an hyperbola having its 

 asymptotes coinciding with the axes. 



8. Find the locus of the ultimate intersections of the 

 straight lines drawn at right angles to normals to 

 the parabola y 3 = 4a#, at the points where they cut 

 the axis. 



Result, y 2 = 4a (2a x}. 



