148 BOOK OF MATHEMATICAL PROBLEMS. 



where < is the angle which the tangent at the point of contact 

 makes with the axis. 



724. The locus of the centre of a rectangular hyperbola 



x* ?/* 



having contact of the third order with the ellipse _ + r , = 1 is 



a" b 



the curve 



725. The locus of a point, such that tangents drawn from it 

 to a fixed conic intercept a constant length on a fixed tangent, is a 

 conic having contact of the third order with the given conic at the 

 extremity of the diameter through the fixed point of contact ; and 

 that the locus is an ellipse, parabola, or hyperbola, according as 

 the given length is less than, equal to, or greater than the diame- 

 ter parallel to the fixed tangent. 



726. On every straight line can be found two points, conju- 

 gate to each other with respect to a given conic, and the distance 

 between which subtends a right angle at a given point not on the 

 straight line. 



727. If a parabola meet an ellipse in four points such that 

 the four normals to the ellipse at those points meet in one point, 

 the axis of the parabola will be parallel to one of the equi-con- 

 jugate diameters of the ellipse. 



728. A conic is drawn through four given points lying on 

 two parallel straight lines ; prove that the asymptotes touch the 

 parabola which touches the other four joining lines. 



