CONIC SECTIONS, ANALYTICAL. 1G3 



779. The radius of curvature at the point B of the conic 

 sin'C 



is 



-j. -- r p . 

 k am A sm.fi 



780. OA, OS touch a conic at A, B, and the tangent at P 

 meets OA, OB in B', A' respectively ; find the locus of the inter- 

 section of AA' t Bff; and if AP, BP meet OB, OA in A", B" 

 respectively, find the envelope of A"ff'. 



781. OA, OB touch a conic at A, B, and C, D are two other 

 fixed points on the conic ; a tangent to the conic meets OA, OB 

 in C', D'] prove that the locus of the intersection of CC', DM 

 is a conic passing through C, D and the intersections of 00, BD; 

 aiid of OD, AC. 



782. The locus of the foci of the conic fiy = ka? for different 

 values of k is 



cot B - cot = 0. 



sin A \sin f B sin* Cj sin B sin 



783. Prove that the equation 4/?y = a* represents a parabola; 

 and that the tangential equation of the same parabola is yz = x*. 



784. CA, CB are tangents to a conic, P any point on the 

 conic, and AP, BP meet CB, CA in A\ B' ; prove that the tri- 

 angle A'B'P is self-conjugate to another conic touching the former 

 at A, B. 



785. The sides of a triangle ABC touch a conic, 0, p 0,, O t 

 are the centres of the circles which touch the sides ; a conic is 



bed through B, C, 0, 0, and one focus, and another through 

 B, C, O t , 9 and the same focus; prove that the fourth point "t 

 intersection of these conies will bo the second focus. 



786. A conic passes tli n.u^li four given points; prove thai 

 the locus of the points of contact of tangents drawn to it from a 

 given point is in general a tin third degree, whi.-h r 



duces to a conic if the point 1"' in tho same straight line with two 

 of the former; and in that case the locus passes through th other 

 two given points. 



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