Elements of Analytical Geometry (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1873 Excerpt: ...arc; the points of intersection will be points of the curve. For, we have F'M--FM = 2a. It may also be constructed by a continuous movement. Take a rule of sufficient length as F'L, and fasten one end at the focus F'; at the other end of the rule fasten one end of a string shorter than the rule by the transverse axis; fasten the other end of the string at the other focus, F; press a pencil against the string and rule; as the rule revolves about the focus F', the point of the pencil will describe the branch AM. For, we have F'L--2a = FM + ML, or F'L--ML-FM = 2a; henco F'M-FM = 2a. By placing the end of the rule at F, the other branch may be described. 122. By a reference to equations (1) and (2), Art. (120), it, is seen that the distance from any point of the curve to either focus is expressed rationally in terms of its abscissa. This remarkable property of the foci is possessed by no other points in the plane of the curve. For, if there is any other point, let its co-ordinates be x' and y'; x and y denoting the co-ordinates of any point of the curve. The square of the distance from x, y, to x', y', Art. (17), is D = (-x'Y + (y-y'Y, or squaring x--x' and y--y', and substituting for y its value --Va-A we have D = a b2 x2--2xx' + x" + Vs =F 2y'--Vau _ x, yi a a 'y It is evident that the value for D can not be rational, in terms of x, unless the term containing the radical disappears. But this can not be unless y' = 0, that is, the required point must be on the axis of X. Substituting this value for y', I), after changing the order of the terms, becomes D2 = (6 + x'2)-2xx' + ffi" x. a% Now no value of x' can make this expression a perfect square unless it makes the first and last terms perfect squares, and twice the square root of their product ...