10- BOOK OP MATHEMATICAL PROBLEMS. 



773. If la + m/2 + ny^Q be the equation of the axis of a 

 parabola which touches the sides of the triangle of reference, 



la* mb* nc* 

 - + - j+-, - =0. 

 in n n l l m 



774. A conic touches the sides of the triangle ABC, any point 

 is taken on the straight line which passes through the intersec- 

 tions of the chords of contact with the corresponding sides, and 

 the lines joining this point to the angular points meet the oppo- 

 site sides in A' t B*, C' ; prove that corresponding sides of the 

 triangles ABC, A'B'C' intersect two and two on one straight line 

 which touches the conic. 



775. If a, b be the axes of a conic inscribed in a triangle, 

 and a, /?, y be the trilinear co-ordinates of either focus : 



I* _ 4a(3y(a sin A + ... + ...) (/?y sin ^1 + ... + ...) 



~~ 



776. Two parabolas are such that triangles can be inscribed 

 in one whose sides touch the other; prove that 



'* 



if (a, /?, y), (a', /?', y) be the foci; the triangle formed by the 

 common tangents being the triangle of reference. 



777. A conic touches the sides of the triangle ABC in the 

 points A', K, C'; A A', BK, CC' meet in 0; through is drawn 

 any straight line meeting the sides of the triangle in a, 6, c, and 

 from a, 6, c are drawn other tangents to the conic ; prove that 

 these tangents will intersect two and two in points lying on a 

 fixed conic circumscribing the triangle. 



778. The equation of an asymptote of the conic (3y = ka*, is 



/n'y + /?-2fyxa=0, 



/A being given by the equation /A* + /* + k = ; also the envelope 

 of the asymptotes for different values of k is the parabola 



