CONIC SECTIONS, ANALYTICAL. 1 -_7 



G23. Two normals are drawn to an ellipse at the extremi- 

 ties of a chord parallel to the tangent at the point a ; prove that 

 the locus of their intersection is the curve 



2 (ax sin a -f by cos a) (ax cos a + by sin a) = (a* 6*) f sin 2a cos 1 2a. 



G24. The normals at three points of an ellipse whose eccen- 

 tric angles are a, /?, y will meet in a point if 



sin (ft + y) + sin (y + a) + sin (a + ft) = 0. 



625. If four normals to an ellipse meet in a point, the sum 



of the corresponding eccentric angles will be an odd multiple 



Prove also that two tangents drawn to the ellipse parallel 



to two chords passing through the four points will intersect on 



one of the equi- conjugate diameters. 



G26. If (tfpy,), (x t , y f ), ( y.) be three points on ah 

 ellipse, such that x t + x a -t- x t = 0, y, + y f + y a = 0, the circles of 

 curvature at these points will pass through a point on the ellipse, 



whose co-ordinates are l -f-^ , y r ' 

 a o 



627. If four normals be drawn from a given point to any one 

 of a series of confocal ellipses, the sum of the angles made by 

 them with the axis is constant 



* 



3, The normals to an ellipse at the points where it is met 



by the straight lines 



- 1 

 + _. 



:i intersect in a point 



Pp is a diameter of an ellipse, PJ/ t PJV perpendiculars 

 II the axes, MN produced meets the ellipse in Q, q\ prove that 

 the normals at Q, q, intersect in the centre of curvature at p. 



630. If from a point be drawn OP, OQ, OR, OS normals to 

 an ( -11 ip.se, and p, q, r t 9 be taken such that their co-ordinates arc 



