ENVELOPS. EXA1VIPLES. 365 



9. Straight lines drawn at right angles to the tangents 

 of a parabola at the points where they meet a given 

 straight line perpendicular to the axis, are in general 

 tangents to a confocal parabola. 



10. Find the envelop of the curves ( y ) + ( JL~~) = *> 



the variable parameters a, b, being connected by the 

 b 



A . /a 

 equation^ 



a- 8 ?/ 2 



Result. ^ + r< = 



11. Circles are described on successive double ordinates of a 



parabola as diameters : shew that their envelop is an 

 equal parabola. Find what part of this system of 

 circles does not admit of an envelop. 



12. A circle moves with its centre on a parabola whose 



equation is y* 4ax = 0, and always passes through 

 the vertex of the parabola : shew that the circle always 

 touches the curve y* (x + 2a) + x 3 = 0. 



13. A series of parabolas of latus rectum I is described with 



their vertices in a given parabola of latus rectum I'. 

 Shew that the locus of the ultimate intersections is a 

 parabola with latus rectum I + 1', the concavities being 

 in the same direction and the axes parallel. 



14. Find the envelop of all ellipses having the same centre 



and in which the straight line joining the ends of the 

 axes is of constant length. 



fiesult. xy = c. 



x* v* 



15. From any point of the ellipse -5 + j^ = 1, perpendiculars 



are drawn to the axes, and the feet of these perpen- 

 diculars are joined: shew that the straight line thus 



formed always touches the curve (-) + [ r 1 = 1. 



w W 



