90 BOOK OF MATHEMATICAL PROBLEMS. 



427. If on a tangent to a parabola be taken two points equi- 

 distant from the focus, the two other tangents drawn to the para- 

 bola from these points will intersect on the axis. 



428. A circle is described on the latus rectum of a parabola 

 as diameter, and a straight line drawn through the focus meets 

 the curves in P,Q', prove that the tangents at P, Q intersect 

 either on the latus rectum, or on a straight line parallel to the 

 latus rectum at a distance from it equal to the latus rectum. 



429. A chord of a parabola is drawn parallel to a given 

 straight line and on this chord as diameter a circle is described ; 

 prove that the distance between the middle points of this chord 

 and of the chord joining the other two points of intersection 

 of the circle and parabola is of constant length. 



430. On any chord of a parabola as diameter is described a 

 circle cutting the parabola again in two points ; if these points be 

 joined the portion of the axis of the parabola intercepted between 

 the two chords is equal to the latus rectum. 



431. A parabola is described having its focus on the arc, its 

 axis parallel to the axis, and touching the directrix, of a given 

 parabola ; prove that the two curves will touch each other. 



432. Circles are described having for diameters a series of 

 parallel chords of a given parabola; prove that they will all 

 touch another parabola related to the given one in the manner 

 described in the last question. 



433. The locus of the centre of the circle circumscribing the 

 triangle formed by two fixed tangents to a parabola and any other 

 tangent is a straight line. 



434. Two equal parabolas, A and B, have a common vertex 

 and axes in the same straight line ; prove that the locus of the 

 poles with respect to B of tangents to A is A. 



435. Three common tangents PP f t QQ' t RR are drawn to 

 two parabolas and PQ, P'Q' intersect in L ; prove that LR, LR' 

 are parallel to the axes of the two parabolas. 



