CONIC SECTIONS, ANALYTICAL. 171 



817. If A, B, C, A', 2?, C" be six fixed points on a conic, such 

 that 



and P, F be any two points of the conic, such that 



PP will pass through a fixed point on BB'. 



818. A given straight line meets any conic which passes 

 through four given points in two points ; prove that these points 

 are conjugate with respect to the conic, which is the loens of the 

 pole of the given straight line with respect to the series of conies 

 through the four points. 



819. The anharmonic ratio of the pencil subtended by the 

 four points a t , a f , a gt a 4 on an ellipse at any point on the ellipse is 



sin ' * sin 



L 



2 2 



820. If tangents be drawn to a conic at A, 7?, C f , D, and 

 -V |t -V f , X t be the middle points of the diagonals joining the points 

 of intd--. tion of the tangents at (1) A, JB-, C, J); (2) A, C '; 

 7?, D; (3) A, I); B, C \ and bo the centre; the range 

 {OJTjJTgJr,} is equal to the pencil {ABCD} at any point of the 



821. A conic is drawn through four jivm points -4, B, C, D; 

 AB, CD meet in }' AC, Bl> in )' f , AD, BC in K a , and is the 



I>p>vr tint tli.- pencil {.-I BCD} on the c.un'c 



is equal to the pencil \O}\Y t Y A \ on the o.nir which is the I.TUS 

 of 0. 



822. The anharmonic ratio of the points of intersection of 



the conies 



fa > +m/3 f + n/-0, fa' + m'/? f + n'/ = 0, 



