CONIC SECTIONS, ANALYTICAL. 14? 



718. The ellipse of least eccentricity which can be inscribed 

 in a given parallelogram is such that any pciiit of contact divides 

 the side on which it lies into parts which are to each other as 

 the squares on the adjacent diagonals. 



719. The axes of the conic which is the locus of the centres of 

 all conies passing through four given points are parallel to the 

 asymptotes of the rectangular hyperbola which passes through the 

 four points. 



720. The equation of a conic having the centre of the ellipse 

 -5 -f ~i = 1 for focus, and osculating the ellipse at the point 0, is 



= {(a" - 6 1 ) (ax cos 8 - by sin 8 0) + a'&'}'. 



7-1. A parabola has contact of the third order with nn 

 ellipse ; prove that the axis of the parabola is parallel to tho 

 diameter of the ellipse through the point of contact, and that the 



latus rectum of the parabola is equal to -^ - , where a, b are the 

 semiaxes and r the central distance of the point of contact. 



_\ The locus of the foci of all conies which have a contact 

 of th- third order with a given curve at a givt n |..,int, is a rui\t> 

 whoa- 1 to the tangent and normal at the given 



I ic form 



- 2a) = ay (a: - a). 



'. An ellipse of constant area ire* is descriU.l having 



contact of tho third h a given parabola whose latus 



: prove that the locus ..f of tho ellips.- 



rxl parabola whose vertex is at a distance f J tY.un tho 



vertex of the given parabola ; also that, when c = m, the axes of 

 the ellipse make with the axis of the parabola angles 



? tan- (2 tun *), 



102 ' 



