120 BOOK OF MATHEMATICAL PROBLEMS. 



, lx ///>/ , I'x m'y 



Now if h -ry = 1. and 1 , - = 1. be the equations of 



a b a b 



two lines joining the four points, the equation 



a*^b> \a ' b V\ & 



can be made to coincide with (A). This leads to 



X = 1, #'+1=0, mm +1 = 0, 



whence we see that the normals at the points where the ellipse is 

 met by the straight lines 



lx my x y _ 



T J~ 1. * T -f JL 



a b al bin 



meet in -a point. If a, /? be the two points on the former line, 

 and y one of the points on the other, 



a+ 8 a+ ft 



cos ~ sin : 



2 2 cos y sin y 



cos ^- cos -^- 



whence 



sin -- -- cos - + (cos y sin - +siny cos jr 1 cos - = 0, 



4 *A \ A & / 



or sin (/2 + y) + sin (y + a) + sin (a + /3) = 0. 



The same relation must hold for the fourth point (8), whence 

 the other equation 



immediately follows. 



589. A is the vertex of an ellipse of eccentricity e, along PA 



PA 

 any chord is taken a length PR = , and RQ is drawn at right 



angles to the chord to meet the straight line through P parallel to 

 the axis ; prove that the locus of Q is a straight line perpendicu- 

 lar to the axis. 



590. Two ellipses have the same major axis, and an ordinate 

 NPQ is drawn ; the tangent at P meets the other ellipse in two 



