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140. Some properties of the parabola involving diameters. 



Illation of thu diainctrr ol' the parabola (Art. 1 



IT -, ... (.) 



shmvs at once tliiit crcrif ///////// nf tin 1 jmr J 



to the axis of the curve. (See also Kx. 8, p. 213.) 



Conversely, since any value whatever may be assign << I 

 to wt, each value determining a system of parallel (holds, 

 equation (1) may represent any lino parallel to tin; a*-a\is, 

 and t heivf< ire every line parallel to ///< .ris of a parabola bisects 

 some set of parallel chords, and is a diameter of the curve. 



Again, each of the chords cuts the paral>ola in general in 

 two distinct points, ami t lie nearer these chords are to tlio 

 extremity of the diameter the nearer are theso two points 

 to each other and to their mid-point. In the limiting ]><>M- 

 tion, when the chord passes through the extremity of tho 

 diameter, tlie two intersection points and tlieir mid-point 

 heco me coincident, and tlio chord is ;L taii'_rent. There 

 the tanyent at the end of a diameter is parallel to the bisected 

 chords. 



It follows from the preceding properties, <r directly from 

 equation (1), that the axis of the parabola is the <>///// <//'<// 

 perpendicular to the t <!//>/ /if, at its extremity. 



The student will readily perceive how the. nl>uve properties 

 give a method for constructing a diameter to a set of rhords, 

 and in particular how to construct the axis of a given parah- 

 ola. Thus the prol.lem of Art. loT, to construct a tangent 

 and normal to a given para hoi a at a given point, can nov. 

 s< dved even when the axis is not given. 



If any point on a diameter is taken as a pole, its polar 

 will be one of the system of bisected chords, of slope in. 



