112 BOOK OF MATHEMATICAL PROBLEMS. 



566. A straight line parallel to the directrix of a parabola 

 meets the axis produced at a distance from the vertex equal to 

 the latus rectum, and a point P on this straight line is joined to 

 the vertex A by a straight line meeting the directrix in Q ; with 

 centre Q and radius QA is described a circle meeting the pa- 

 rabola again in R; prove that PR will be normal to the parabola 

 at R. 



567. A chord of a parabola passes through the centre of 

 curvature at the vertex ; prove that the normals to the parabola 

 at the extremities of the chord intersect on the parabola. 



568. From any point of a straight line drawn through the 

 focus of a parabola, and making an angle a with the axis, three 

 normals are drawn ; prove that the sum of the angles which they 

 make with the axis exceeds a by a multiple of TT. 



569. Normals are drawn at the extremities of any chord 

 passing through a fixed point on the axis of a parabola; prove 

 that their point of intersection lies on a fixed parabola. 



570. Two normals to a parabola meet at right angles, and 

 from the foot of the perpendicular let fall from their point of 

 intersection on the axis, is measured towards the vertex a distance 

 equal to one fourth of the latus rectum ; prove that the straight 

 line joining the end of this distance with the point of intersection 

 of the normals will also be a normal. 



571. Two equal parabolas have their axes coincident but 

 their vertices separated by a distance equal to the latus rectum ; 

 through the centres of curvature at the vertices are drawn chords 

 PQ, l y Q', equally inclined in opposite directions to the axis, 

 P, P' being on the same side of the axis ; prove that (1) PQ\ 

 PQ are normals to the outer parabola; (2) their intersection 

 JZ lies on the inner parabola; (3) the normals to the inner 

 parabola at P f ) Q', K meet in a point which lies on a third 

 equal parabola. 



