

CONIC SECTIONS, GEOMETRICAL. 93 



the parabola and equal to half the latus rectum; prove that it will 

 touch the parabola. 



452. In a conic PG is the normal at P, S the focus ; prove 

 that if SG = PG, SP is equal to the latus rectum. 



453. CPQ is a common radius to the circles on the minor 

 and major axes of an ellipse, and tangents to the circles at P, 

 Q meet these axes in U, T j prove that TU will touch the ellipse. 



454. S, S' are the foci of a conic, SPY, S'P'Y' perpendicu- 

 lars on a tangent to the auxiliary circle meeting the conic in P, 

 r ; prove that the rectangle SY y &P'=tlie rectangle T', 

 SP = BC\ 







455. Given the foci and the length of the major axis; obtain 



by a geometrical construction the points in which the conic meets 

 a given straight line drawn through one of the foci. 



450. A tangent to a conic at P meets the minor axis in T, 

 and TQ is drawn perpendicular to SP one of the focal distances ; 

 prove tli&tSQis of constant length : and, 7M/ l>eing drawn perpen- 

 dicular to the minor axis, that QM will pass through a fixed 

 point. 



457. One focus of a conic, a tangent line, and the length of 

 the major axis is given; prove that the locus of the second focns 

 is a circle. Determine the portions of the locus which correspond 

 to an ellipse, and to a hyperbola of which tin- ijivrn point is an 

 interior focus to the branch touched by the given straight line, 



458. PCG is a diameter of a conic, QV(? a parallel chord 

 bisected in P, PV intersecto CQ, or CQ' in -R; prove that the 

 locus of R is a parabola. 



.'. lfCP,CD be conjugate mdii of an ellipse, and if through 

 C a straight line be drawn parallel to either focal distance 1 

 UK- distance of D from this straight line will be equal to half the 

 minor axu. 



