CONIC SECTIONS, ANALYTICAL. 105 



793. If a triangle be self-conjugate to a parabola, the lim-.s 

 joining the middle points of the sides are tangents to the parabola. 

 Hence prove that the focus lies on the Nine Points' Circle, and 

 that the directrix passes through the centre of the circumscribed 

 circle. 



794. A conic is drawn touching the four straight lines 



la * mfi * ny = ; 

 prove that its equation is 



Z, J/, N satisfying the equation 



Investigate the species of this conic with reference to the position 

 of its centre on the straight line which is its locus, 



795. Any conic through the four points 



la = W/J = *= ny 

 will divide harmonically the straight line joining the points 



K. A. y t )> (<v & y,) if 



796. If a triangle be self-conjugate to a rectangular hyper- 

 bola, and any conic be inscribed in the triangle, its foci will be 

 conjugate points with respect to the hyperbola. 



797. A given triangle is self-conjugate to a conic, and tlm 

 centre of the conic lies on a given straight line parallel to one of 

 the sides of the triangle ; prove that the asymptotes will envelope 

 a fixed conic which touches the other two sides of the triangle. 



798. The locus of the foci of all conies touching the four 

 straight lines la * m/? * ny - is 



(a + p + y) (JV cot A + m'/P cot B + n*/ cot C) sin A sin B sin C 



= (Pa + m'/3 + n'y) (0y sin' A + ya sin' B + aft sin' C). 



