

CONIC SECTIONS, ANALYTICAL. 179 



conies, the polar of the given point with respect to it will ! 

 a tangent to the conic which has the given point for focus and 

 touches the sides of the triangle, and the conic A will subtend 

 a right angle at the given point. 



836. Prove that with the centre of the circumscribed circle 

 as focus, three hyperbolas can be described, passing through the 

 angular points of the triangle ABC ; that their eccentricities are 



B cosec C, kc. ; their directrices the lines joining the middle 

 points of the sides; and that the fourth point of intersection of 

 any two lies on the straight line joining one of the angles to the 

 middle point of the opposite side. 



837. From a point P on the circle circumscribing a triangle 

 ABC, are drawn PA', PR, PC at right angles to PA, PB, PC to 

 meet the corresponding sides ; prove that the straight line A'B'C' 

 passes through the centre of the circle. 



838. A triangle is inscribed in an ellipse so that the centre 

 of the inscribed circle coincides with one of the foci ; prove that 



the radius of the inscribed circle is : - ^ - ^ ; 2c being the 



latus rectum and e the eccentricity. 



^ 

 e ) 



839. A triangle is self- conjugate to a hyperbola, and one 

 focus is equidistant from the sides of the triangle; prove that each 



distance is -77-= - , 2c being the latus rectum and e the ec- 



w - *) 



centricity. 



840. Two conies have a common focus, and are such that 

 triangles can be inscribed in one which are self-conjugate to the 



othrr ; pi-ovi- that 



2c, f + c. f - e/c/ + e,V - 2, Vl c, cos o ; 



c,, c, being their later* recta ; ,, f their eccentricities ; and a 

 the angle between their axes. Prove also that in this en 



it be circpmsci ibed to the second, which arc self-con- 

 jugate to the first. 



