CONIC SECTIONS, GEOMETRICAL. 



I. Parabola. 



411. Two parabolas having the same focus intersect; prove 

 that the angles between their tangents at the points of intersec- 

 tion are either equal or supplementary. 



412. A chord PQ of a parabola is a normal at P and sub- 

 tends a right angle at the focus S j prove that SQ is twice SP. 



413. A chord PQ of a parabola normal at P subtends a right 

 angle at the vertex ; prove that SQ is three times SP. 



414. Two circles each touch a parabola and touch each other 

 at the focus of the parabola ; prove that the angle between the 

 focal distances of the points of contact with the parabola = 120. 



415. Two parabolas have a common focus and their axes in 

 opposite directions ; prove that if a circle be drawn through the 

 focus touching both the parabolas the line joining the points of 

 contact subtends at the focus an angle of 60. 



416. In a parabola AQ is drawn through the vertex A at 

 right angles to a chord AP to meet the diameter through P in Q- } 

 prove that Q lies on a fixed straight line. 



417. Through any point P of a parabola a straight line 

 QPy is drawn perpendicular to the axis and terminated by the 

 tangents at the extremities of the latus rectum ; prove that the 

 distance of P from the latus rectum is a mean proportional be- 

 tween QP, P&. 



418. The locus of a point dividing in a given ratio a chord of 

 a parabola which is parallel to a given line is a parabola. 



419. From any point on the tangent at any point of a para- 

 bola perpendiculars are let fall on the focal distance and on the 



