Elements of analytical geometry and the differential and integral calculus . point on the curve corresponding todefinition 1. Hence, FV=VH. If the square be turned over and moved in the opposite di-rection, the other part of the parabola, the other side of the lineFIImsLy be described. 3. A diameter to a parabola is a straight line drawn throughany point of the curve perpendicular to the directrix. Thus, theline HF\s a diameter ; also, ^6^^ is a diameter ; and all diame-ters are parallel to one another. 4. The point in which the diameter cuts the curve, is calledthe vertex. 5. The axis of the
Elements of analytical geometry and the differential and integral calculus . point on the curve corresponding todefinition 1. Hence, FV=VH. If the square be turned over and moved in the opposite di-rection, the other part of the parabola, the other side of the lineFIImsLy be described. 3. A diameter to a parabola is a straight line drawn throughany point of the curve perpendicular to the directrix. Thus, theline HF\s a diameter ; also, ^6^^ is a diameter ; and all diame-ters are parallel to one another. 4. The point in which the diameter cuts the curve, is calledthe vertex. 5. The axis of the parabola is the diameter which passesthrough the focus. 6. The parameter to any diameter is the double ordinate whichpasses through the focus. 7. The parameter to the principal diameter is sometimes calledthe latus-rectum. PROPOSITION I. To find the equation of the curve. The vertex of the parabola is the zeropoint, or the origin of the co-ordinates. The distance of the focus F, in the direc-tion perpedicular to BH, is called ^, aconstant quantity, and when this constant. 66 ANALYTICAL GEOMETRY. is large, we have a parabola on a large scale, and when small, wehave a parabola on a small scale. By the definition of the curve, V is midway between F andthe line BH, and PF=PB, Put VD=x and PD=y, and operate on the right angledtriangle PDF. FD=x—\p, PB=x-\-\p—PF. (FDy-\-(PDy=^{PFy. That is, {x—^pY+y={x-\-lpY. Whence ^*=2pa;, the equation sought. Corollary 1. If we make x=0, we have y=0 at the sametime, showing that the curve passes through the point F, cor-responding to the definition of the curve. As y=rh^2pa;, it follows that for every value of x there aretwo values of y, numencally equal, one -|-, the other —, whichshows that the curve is symmetrical in respect to the axis of X. Corollary 2. If we convert the equation (y^=2px) into aproportion, we shall have X : y : : y : 22>, a proportion showing that the parameter of the axis is a third pro-portional to any abscissa a
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