230 AX A i > //< 0j wjrrjM L i\. 



5. Die polars of all points on th<> l;itu> r.-.-hnn m.-.-t the axis of tlm 

 parabola in the same point; find its coordinates, for the parabola 



*=. 



6. Phe product of the segments of any focal chord of the pa 

 y* = 4px equals p times the length of the chord. 



7. Two tangents are drawn from an external point, P l = (r } , ?/i) to 

 a parabola, and a third is draun parallel to ih.-ir chord of contact, 

 intersection of the third with each of the other two is half way bet 



P! and the corresponding point of contact. 



8. The area of a triangle formed by three tangents to a parabola is 

 one half the area of the triangle formed by the three points of tanp 



9. The tangent at any point of the parabola will meet the dii 

 and latus rectum produced, in two points equidistant from the focus. 



10. The normal at one extremity of the latus rectum of a parabola is 

 parallel to the tangent at the other extremity. 



11. The tangents at the ends of the latus rectum are twice as far 

 from the focus as they are from the vertex. 



12. The circle on any focal radius as diameter touches the tangent 

 drawn at the vertex of the parabola. 



13. The line joining the focus to the pole of a chord bisects the 

 subtended at the focus by the chord. 



14. Prove, geometrically, that a perpendicular let fall from the focus 

 upon a tangent line of a parabola meets that tangent upon the tangent 

 drawn at the vertex (cf. (7) of Art. 13H. ].. 



139. Diameters. A diameter has been defined as tl 

 hx-us of the middle points of a system of parallel chordi 

 \\< -<| nation may be found as follows (cf. Art. 129): 



Let m be the common slope of a system of parallel choi 

 of the parabola whose equation is 



y* = 4px, . . (1 



tiien the equation of one of these chords is 



(2) 



