CONIC SECTIONS, ANALYTICAL. 181 



the point of contact will lie on the conic touching the sides of the 

 triangle at the feet of the perpendiculars. 



848. A conic is inscribed in a triangle, and one directrix 

 passes through the centre of perpendiculars ; prove that the 

 corresponding focus lies on the circle to which the triangle is 

 self-conjugate. 



849. With the centre of the circumscribing circle of a tri- 

 angle as focus are described two ellipses, one touching the sides of 

 the triangle, and the other passing through the middle points ; 

 prove that these will touch each other. 



850. Five points are taken, no three of which lie in one 

 straight line, and with one of the points as focus are described 

 four conies, each of which touches the sides of a triangle formed 

 by joining three of the other points ; prove that these conies will 

 h;ivo a common tangent. 



851. Through a fixed point are drawn any two straight lines 

 meeting a given conic in P, P f Q, (X; and a given straight line 

 in J?, 7* ; and L'L" subtends a right angle at another fixed point 

 Prove that PQ, PQ> P'Q, FQ all touch a certain fixed conic. 



852. Given a conic and a point in its plane ; prove that 

 there exist two points L, such that if any straight line through // 



hr jM.lur of L in /', and I" be the pole of this straight line, 

 PP subtends a right angle at 0. 



853. Tin- envelope of a straight line which is divided har- 

 monically by two given straight lines and a given c.>nie, 

 conic touching the two fixed straight lines at points on the g 



of th. ntersection ; unless the given lines are conju- 



gate with respect to the conic, when only one such straight lino 

 can In- drawn. 



854. Two equal circles J, /.', t>urh at S, a tangent to B 

 meets A in /', Q. and <> i III j*"le with respect to A ; prove that 

 the di two of the conies described with focus S to 



be triangle O/'V will t.-urh A. 



