CONIC SECTIONS, GEOMETRICAL. 



460. Qf/ is a chord of an ellipse parallel to one of the equi- 

 conjugate diameters, QN, QN' are drawn perpendicular to the 

 major axis; prove that the triangles QCN, Q'CN' are equal ; also 

 that the normals at QQ intersect on the diameter which is per- 

 pendicular to the other equi-conjugate. 



470. Any ordinate NP of an ellipse is produced to meet the 

 auxiliary circle in <?, and normals to the ellipse and circle at P, Q 



in. R; RK, RL are drawn perpendicular to the axes; prove 

 that K, 1\ L lie on one straight line and that A'/*, PL are equal 

 respectively to the semi axes. 



471. Two conies are described having a common minor axis, 

 and such that the outer touches the directrices of the inner; 

 MPP 1 is a common ordinate; prove that MP is equal to the 

 normal at P. 



' i >PP r drawn perpendicular to the major axis of an 

 ell i i -so meets the ellipse in P, P and the auxiliary circle in Q ; 

 prove that the part of the normal to the circle at Q intercepted 

 between the normals to the ellipse at P and P' is equal to the 

 minor axis. 



473. The perpendicular from the focus of a conic on any 

 tangent and the central radius to the point of contact will in- 

 tersect on the directrix. 



L < >n the normal to an ellipse at P are taken two points 

 , such that QP = (?P = GD; prove tluu the mrim of the 



a g ,o 



475. A hyperbola is do h the focus of a paro- 

 tid with its fori lying on the parabola; prove that one < 



asymptotes is parallel to the axis of the parabola, 



476. A parabola passes through two given points and its 

 |..ir.il!.-l to a given lin that the locus of its focus is 



a hyperbola. 



