126 BOOK OF MATHEMATICAL PROBLEMS. 



619. Two tangents OP, OQ are drawn at the points a, /?; 

 prove that the co-ordinates of the centre of the circle circumscrib- 

 ing the triangle OPQ are 



a + J3 



cos 



2 a 2 + (a?-b*) cos a cos/3 



cos 



' 



2 6* + (6* -a*) sin a sin 



26 



2~ 



If this point lie on the axis of x, the locus of is a circle. 



620. Two points P, $ are taken on an ellipse, such that the 

 perpendiculars from Q, P on the tangents at P, Q intersect on the 

 ellipse j prove that the locus of the pole of PQ is the ellipse 



If R be another point similarly related to P, the same relation 

 will hold between Q, R : and the centre of perpendiculars of the 

 triangle PQR will be the centre of the ellipse. 



621. Perpendiculars p } , p 2 are let fall from extremities of two 

 conjugate diameters on any tangent, and p a is the perpendicular 

 from the pole of the line joining the two former points : prove 

 that p* = Zp^pf 



622. The normal at a point P of an ellipse meets the curve 

 in Q, and any other chord PP' is drawn; $P'and the line through 

 P at right angles to PP' meet in R : prove that the locus of R 

 is the straight line 



x y . 



< being the eccentric angle of P. The part of any tangent inter- 

 cepted between this straight line and the tangent at P will be 

 divided by the point of contact into two parts which subtend 

 equal or supplementary angles at P. 



