. Mathematics, compiled from the best authors and intended to be the text-book of the course of private lectures on these sciences in the University at Cambridge [microform]. Cor. 3. Hence the tangents to all parabolas, whichhave the same abscisses, meet the axis produced in thesame point. For, if the absciss AM be the same in all,the external axis AI, which is equal to it, will be the samealso. Cor. 4. And hence also a tangent is easily drawn tothe curve. For, if the point of contact C be given, draw the ordin-ate CM, and produce MA till AI be=^Alvi ; then joinIC the tangent. Or, if the point
. Mathematics, compiled from the best authors and intended to be the text-book of the course of private lectures on these sciences in the University at Cambridge [microform]. Cor. 3. Hence the tangents to all parabolas, whichhave the same abscisses, meet the axis produced in thesame point. For, if the absciss AM be the same in all,the external axis AI, which is equal to it, will be the samealso. Cor. 4. And hence also a tangent is easily drawn tothe curve. For, if the point of contact C be given, draw the ordin-ate CM, and produce MA till AI be=^Alvi ; then joinIC the tangent. Or, if the point I be given, take AM = AI, and drawthe ordinate MC, which will give the point of contact C,to which draw IC the tangent. 128 MATHEMATICS. PROPOSITION VI. If a tangent to the curve meet the axis produced, thenthe line drawn from the focus to the point of contact,will be equal to the distance of the focus from the intersection of the tangent and axis. That is, FC = For, draw the ordinate DC to the point of , by Prop. V. Cor. 2, AT=AD;therefore, FT = A F -f A D. But, by Prop. IV. FC = AF + AD ; therefore, by equality, FC = FT Q. E. D« Cor. 1. If CG be drawn perpendicular to the curve,or to the tangent at C, then shall FG — FC = FT. For, draw FH perpendicular to TC, which will also bi-sect TC, because FT=FC ; and therefore, by the natureof the parallels, FH also bisects TG in F. And conse-quently FG = FT = FC. So that F is the centre of a circle passing through T?C, G. Coit. 2. The subnormal DG is every where equal t»the constant quantity 2FA, or \ P, the semiparameter. CONIC SECTIONS. 329 For, draw the tangent AH parallel to DC, making thetriangle FHA similar to GCD. Then DC = 2AH, because DT = 2DA ;consequently DG = 2FA = | P. Cor. 3. The tangent at the vertex AH is a mean pro-portional between AF and AD. For, because FHT is a right angle,therefore, AH is a mean between AF, AT, or between AF, AD, because AD = AT. Likewise FH is a
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