418 MISCELLANEOUS EXAMPLES. 



11. Find the envelop of the axis of a parabola having a 



focal chord given in position and magnitude. 



Result, a? + y* = c* ; the origin being the middle 

 point of the given chord, and one of the axes coinciding 

 with that chord. 



12. A system of ellipses is described such that each ellipse 



touches two rectangular axes, to which its axes are 

 parallel, and that the rectangle under the axes of 

 the ellipse is constant: shew that each ellipse is 

 touched by two rectangular hyperbolas, the rectangle 

 under the transverse axes of which is equal to the 

 rectangle under the axes of any one of the ellipses. 



13. A, B, are the centres of two equal circles, and AP, BQ, 



are two radii which are always perpendicular to each 

 other : find the curve which is always touched by the 

 right line PQ, and explain the result when 



14. Trace the following curves : 

 # 3 - xy* + ay* = 0, 

 y 3 - lyx* + Gx 3 - a 3 = 0, 



a (x 3 + 1x-y + Ixy* + y z ] - afy 9 = 0, 



ary* + ax* - a 3 = 0, 



if (x - 2a) - x 3 + a 5 = 0, 



y 5 ax s y - bxy* + x 6 = 0, 



a ~ a 



y z (a + x}=^(a-x), 



t/ J 



