96 BOOK OF MATHEMATICAL PROBLEMS. 



477. If two tangents of a hyperbola be the asymptotes of 

 another hyperbola and that other touch one of the asymptotes of 

 the former it will touch both. 



478. Two similar conies A, B are placed with their major 

 axes in the same straight line, and the focus of A is the centre of 

 B ; if a common tangent be drawn the focal distance of its point 

 of contact with A will be equal to the semi major axis of . 



479. A series of similar conies are described having the same 

 focus and direction of major axis, and tangents are drawn to them 

 at points where they meet a fixed circle having its centre at the 

 common focus ; prove that these tangents will all touch a similar 

 fixed conic whose major axis is a diameter of the circle. 



480. If a chord of a conic subtend a right angle at each of the 

 foci, it must be either parallel to the major axis or a diameter. 



481. From the foci S, S' of an ellipse perpendiculars SY, 

 S'Y' are let fall on any tangent ; prove that the perimeter of the 

 quadrilateral SYTS' will be the greatest possible when YY' 

 subtends a right angle at the centre. 



482. The angle which a diameter of an ellipse subtends at 

 the extremity of the axis major is supplementary to that which its 

 conjugate subtends at the extremity of the axis minor, 



483. From the focus of an ellipse is drawn a straight line 

 perpendicular to the tangent at a point of the auxiliary circle ; 

 prove that this perpendicular is equal to the focal distance of the 

 corresponding point of the ellipse. 



484. If on any tangent to a conic be taken two points equi- 

 distant from one focus and subtending a right angle at the other 

 focus ; their distance from the former focus is constant. 



485. If a conic be described having one side of a triangle for 

 directrix, the opposite angle for centre, and the centre of perpen- 

 diculars for focus ; the sides of the triangle which meet in the 

 centre will be conjugate diameters. 



