THE CONIC SECTIONS.
By Prof. C.W. MACCORD, Sc.D.
In Fig. 1 let D be a given point, and O the center of a given circle, whose diameter is FG. Bisect DF at A. Also about D describe an arc with any radius DP greater than DA, and about O another arc with a radius OP = DP + FO, intersecting the first arc at P, then draw PD, and also PO, cutting the circumference of the given circle in L. Since PD = PL, and DA = AF, it is evident that by repeating this process we shall construct a curve PAR, which satisfies the condition that every point in it is equally distant from a given point and from the circumference of a given circle. Since PO-PD = LO, and AO-AD = FO, this curve is one branch of the hyperbola of which D and O are the foci.
Bisect DG at B, then about D describe an arc with any radius DQ greater than DB, and about O another are with radius OQ = DQ-FO; draw from Q the intersections of these arcs, the line QD, and also QO, producing the latter to cut the circumference in E. By this process we may construct the curve QBZ, each point of which is also equally distant from the given point D, and from the concave instead of the convex arc of the given circumference. The difference between QD and QO being constant and equal to FO, and AB being also equal to FO, this curve is the other branch of the same hyperbola, whose major axis is equal to the radius of the given circle.
The tangent at P bisects the angle DPL, and is perpendicular to DL, which it bisects at a point I on the circumference of the circle whose diameter is AB, the major axis, the center being C, the middle point of DO. As P recedes from A, it is evident that the angles P D L, P L D, will increase, until DL assumes the position D T tangent to the given circle, when they will become right angles. P will therefore be infinitely remote, and the point I having then reached t, where D T touches the smaller circle, C t S will be an asymptote to the curve. This shows that the measurements from the convex arc, for the construction of A P, are made only from the portion FT of the given circumference.
In the diagram the point Q is so chosen that DL produced passes through E, so that QJ, the tangent at Q, is parallel to PI. It will thus be seen that the measurements from the concave arc, for the construction of BQ, are confined to the portion G T of the given circumference. As DLE rises, the points P and Q recede from A and B, the points L and E approach each other, finally coinciding at T; at this instant I and J fall together at t, so that S S is the common asymptote to A P and B Q.
In Fig. 2 the given point D lies within the circumference of the given circle. Bisect DF at A, and DG at B; about D describe an arc with any radius DP greater than DA, and about O another, with radius OP = OF - DP, these arcs intersect in P, and producing OP to cut the circumference in L, we have PD = PL. Similarly ED = EH, UD = UW, etc. And since PD + PO = LP + PO, DE + EO = HE + EO, and so on, the curve is obviously the ellipse of which the foci are D and O, and the major axis is AB = FO, the radius of the given circle.