124) BOOK OF MATHEMATICAL PROBLEMS. 



607. If two sides of a triangle be given in position and the 

 third in magnitude, the locus of the centre of the Nine Points' 

 Circle of the triangle is an ellipse ; which reduces to a limited 

 straight line if the acute angle between the given sides be 60. 

 If c be the given side, and 2<x the given angle, the axes of the 



c sin 3a c cos 3a 



ellipse are equal to . . , , -. -. -j . 

 4 sin a cos a 4 sin a cos a 



608. If PM, PN be perpendiculars from any point of an 

 ellipse on the axes, and the tangent at P meet the equi-conjugates 

 in Q, R ; the tangents from <2, R to the ellipse will be parallel 

 to MN. 



609. If a right-angled maximum triangle can be inscribed in 

 an ellipse, the eccentric angle of the point at which is the right 

 angle, is 



610. CP, CD are conjugate radii, and PQ is measured along 

 the normal at P equal to m times CD ; prove that the locus of 

 Q is the ellipse 



if. 



*~ > 



(a-mb? (b-ma)* 



PQ being measured inwards or outwards as m is positive or 

 negative. 



611. If a triangle circumscribe a given ellipse, and its centre 

 of gravity lie in the axis of x, at a distance c from the centre, its 

 angular points will lie on the fixed conic 



(*-3c)' y(o?-9c") 

 a 8 2 6 a 



612. If two tangents be drawn to an ellipse from a point P, 

 the cosine of the angle between them is 



