180 BOOK OF MATHEMATICAL PROBLEMS. 



841. Three tangents to a hyperbola are so drawn that the 

 centre of perpendiculars of the triangle formed by them is at one 

 of the foci; prove that the radius of the circle to which the 

 triangle is self-conjugate is constant. 



'.'2. If four points lie on a circle, four parabolas can be 

 described having a common focus, and each touching the side 

 of a triangle formed by joining three of the points. 



843. BE' is the minor axis of an ellipse, and B is the centre 

 of curvature at B '; on the circle of curvature at B is taken any 

 point P, and tangents drawn from P to the ellipse meet the 

 tangent at B in <2, Q* j prove that a conic drawn to touch QB, 

 Q>B' y with its focus at B and directrix passing through B', will 

 touch the circle at P. 



844. If three tangents to a parabola form a triangle ABC, 

 and perpendiculars p lf p a , p 3 be let fall on them from the focus S ; 

 then will 



pj>i sin BSC +p a Pi sin CSA +p^ a sin ASB = 0, 



the angles at S being measured in the same direction. 



845. A triangle ABC circumscribes a parabola, and a;, ?/, z 

 are the perpendiculars from the focus on the sides ; prove that 



sin 2.4 sin 2$ sin 2(7 8 sin A sin B sin C 

 ~^~ ~y*~ ~~7~ ~T~ ~> 



I being the semi latus rectum. 



846. A point S is taken within a triangle, such that the 

 sides subtend at it equal angles, and four conies are described 

 with S as focus passing through A, B, C ; prove that one of these 

 conies will touch the other three, and that the tangent to this 

 conic at A will meet BC in a point A t such that ASA' is a right 

 angle. 



847. With the centre of perpendiculars of a triangle as focus 

 are described two conies, one of which touches the sides of the 

 'triangle, and the other passes through the feet of the perpen- 

 diculars ; prove that these conies will touch each other, and that 



