128 BOOK OF MATHEMATICAL PROBLEMS. 



the intercepts on the axes of the tangents at P, Q, R, S respect- 

 ively ; p, q, r, s will lie on one straight line. Also, if through 

 the centre C be drawn straight lines at right angles to CP, CQ, 

 CRj CS to meet the tangents at P, Q, R, S respectively, the four 

 points so determined will lie on one straight line. 



631. If the normals to an ellipse at P, Q, It, S meet in a 

 point, and circles described about QRS, RSP, SPQ, PQR meet 

 the ellipse again in P', Q', R', S', the normals at P', Q\ #, S' will 

 meet in a point. 



632. PQ is a chord of an ellipse, normal at P, PP' a chord 

 perpendicular to the axis, the tangent at P' meets the axes in T, 

 T' t the rectangle TCT'R is completed, and CR meets PQ in U ' 

 prove that CR.CU = a*- b*. 



633. Normals drawn to an ellipse at the extremity of a 

 chord passing through a given point on the axis will intersect in 

 another ellipse whose axes are 



-&'/ IN a*-l* f 1\ 



- (c + -J, = (c~-) : 

 a V cj b \ c, 1 ' 



the distance of the given point from the centre being c times 

 the semi major axis. 



x s y~ 



634. A triangle A'B'C' is inscribed in the ellipse ^ + ^ 1, 



x 2 -?/* % 



and its sides touch the ellipse + J - t = 1 in the points A, B, C ', 



prove that, if the relation which must exist between the axes be 



-7 + , = 1, the eccentric angles of A and A' will differ by TT : but 

 a b 



if the relation be ~ r . = 1 , the sum of these eccentric angles 

 a o 



will be either v or STT. 



635. Two ellipses are concentric and similarly situated, and 

 triangles can be inscribed and circumscribed ; prove that the 

 normals to one ellipse at the angular points of any such triangle 

 meet in a point ; and also the normals to the other at the points 

 of contact. 



