CONIC SECTIONS, ANALYTICAL. 1 ."> I 



prove that the envelope is the two circles 



ami if the relation between the axes be 



the envelope is py? + qy* + 2aq*x = 0. 



.'. Through each point of the straight line + 



y? V* 

 rawn a chord of the ellipse -, + |j 



prove that the envelope is the parabola 



y? V* 

 is drawn a chord of the ellipse -, + |j = 1 bisected in the point 



733. The envelope of the ellipse 



. fx cos a y sin a\ 



x * + y* - 2 (ax cos a + by sin a) ( + y r ) 



\ o / 



is the two ellipses 



734. A variable tangent to a parabola meets two fixed 

 tangents in two points ; prove that tin- diivr.trix of the parabola 

 wlii.-h touches the fixed tangents at these points envelopes an- 

 parabola. 



7 .J.I. A variable tangent to a parabola meets two fixed tan- 

 , and a circle is described on tin- ]>ait int< r< ptrd as dia- 

 meter; prove that the envelope of these circles is an ellipse 

 \vlnrh touches the fixed tangents at the points where the d 

 meets them. 



73r. I!, director circle of a omiii* :unl one point of the conic 

 being given, provo tint lope is a fixed conic whose maj<r 



axis Is a diameter of the given circle. 



