CONIC SECTIONS, ANALYTICAL. 1 _' '. 



G13. Two points H, II' are conjugate with respect to an 



ellipse, P is any point on the ellipse, and I'll, I'll' meet the 

 ellipse again in QQ' prove that Q(? passes through the pole 

 of ////'. 



G14. A triangle circumscribes the circle of + y* = a', and two 

 of its angular points lie on the circle (a: cf + y* = 6* ; prove that 

 the locus of the third angular point is a conic touching the 

 common tangents of the two circles; and that this conic becomes 

 a parabola if (c * a) 8 = 6' - a*. 



615. A triangle circumscribes an ellipse, and two of it- 

 angular points lie on a confocal ellipse ; prove that the third 

 angular point lies on another confocal ellipse, and that the peri- 

 meter of the triangle is constant. 



GIG. The lines 



form a self-conjugate triangle to the' ellipse ; prove that 



and that the co-ordinates of the centre of perpendiculars < 

 triangle are 



.ingle is self-conjugate to a given ellipse, and one 

 angular point is fixed ; prove that the circle tircumiicr 

 triangle ]>asses through another fixed point Q ; that C, Q t art in 

 one straight line, and that CQ.CO a' + b\ 



618. In the ellipses 



a tangent to the farmer meets the latter in P, Q ; prove that the 

 tangents at /', Q are at right angles to each other. 



