Elements of analytical geometry and the differential and integral calculus . F. (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
Elements of analytical geometry and the differential and integral calculus . F. (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 and its corresponding ordinate. PROPOSITION n. The squares of ordinates to the axis are to one another as their corresponding abscissas. Let X, y, be the co-ordinates of any point P, and x, y\ theco-ordinates of any other point in the curve. Then by the equation of the curve we must have y^ z=i%px, (1) y^=%px\ (2) x\ Q. E. D. By division X Wt lence y •? y^ ;: X :. THE PARABOLA. 67 PROPOSITIOJf III. The lotus-rectum is four times the distance from the focus tothe vertex. Let F VH be a parabola, F the focus, and Vthe principal vertex. FH, at right angles to DF,through the point F, is the latus-rectum. We are to prove that FB=^4FV. In the equation of the curve, (y^=2px) for thepoint F, we must necessarily make x=^p, thenthe equation becomes y=p. That is, FF=FI>=2 VF, or FB=4: VF. Q. E. D. Corollary. It will be observed that CF and DBsire squares,and the line DF or its equal FF is the quantity represented byJ). It is the same for the same parabola, but different in differ-ent parabolas. PROPOSITION W. To find the equation of a tangent line to the parabola. Let the line SPQ cut the parabolain two points P and Q. Let ir, y, be the general co-ordi-nates of any point in the line as S;x\ y the co-ordinates of the point F;and x, y, the co-ordinates of thepoint Q. The equation o
Size: 1228px × 2034px
Photo credit: © The Reading Room / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookauthorrobinson, bookcentury1800, bookdecade1850, bookyear1856
