Page:Encyclopædia Britannica, Ninth Edition, v. 6.djvu/302

Rh 274 CONIC SECTIONS the axis of the parabola and the vertex of the axis is called the principal vertex. COROLLARY. A perpendicular drawn from the focus to the directrix is bisected at the vertex of the axis. A straight line terminated both ways by the parabola., and bisected by a diameter, is called an ordinate to that diameter. The segment of a diameter between its vertex and an ordinate, is called an abscissa. A straight line meeting the parabola in two points P, Q, is called a chord. The focal chord which is bisected by a diameter is called the parameter of that diameter. The limiting position of the chord PQ, which it Assumes when the point Q moves up to and coincides with P. is called the tangent at P. A line through P at right angles to the tangent is called the normal at P. PROPOSITION T. To find where a parabola of given focus and directrix is cut by a straight line parallel to the directrix. Let S (fig. 2) be the focus, and XK the directrix. Draw SX perpen dicular to KX, and bisect SX in A ; draw AZ at right angles to SX, and equal to AS. Join XZ. Let QN be any straight line parallel to the directrix, cutting XZ in Q and the axis in N. With centre S and radius equal to QN, describe a circle cutting QN in P and P ; these will be points on the parabola, because SP : XN = QN- : Xtf - ZA : AX = 1 : I.,. SP = XN = distance of P from the directrix. li, is clear that if the point P Fig. 2. exists, the point P on the opposite side of the axis also exists, and therefore the parabola is symmetrical with respect to the axis. Again, the point P will exist, or, in other words, the circle will cut QN&quot;, as long as SP or QN is greater than SN, which is always the case as long as QN lies on the same side of AZ as the focus. The whole of the curve, therefore, lies to the right of AZ, and branches off to an infinite distance from the directrix. PROP. II. To find where a parabola of given focus and directrix is cut by a straight line parallel to the axis. Let S (fig. 3) Lie the focus, and XK the directrix ; draw AY bisect- I P ing SX at right angles. Let KQ be any line parallel to the axis cutting the directrix in K. Join SK cutting AY in Y, and draw YP at right angles to SK cut ting KQ in P. P will be a point on the curve. It is easily shown that the tri angles SPY, KPY are equal in all respects, and that SP = PK. Fig. 3. Fig. 5. Fig. 4. Now, the point P will exist, or, in other words, YP will inter sect KQ, for all positions of KQ. The parabola, therefore, branches off on either side of the axis to an infinite distance, and is cut by a straight line parallel to tua axis in one point only. It appears from what has gone before that the general sbapo of the curve is of the form given in fig. 4, which shows tho focus, directrix, and axis. It can be easily seen that all points within the curve are nearer to the focus than to the directrix, and all points without the curve are nearer to the directrix than to the focus. A parabola can be described mechanically in the following manner (see fig. 5) : Suppose a bar KQ to move always parallel to itself, with its end K on a line at right angles to it ; then, if a string of length equal to KQ, attached to the bar at Q, anil also to a fixed point S, bo always kept tight by means of a ring P sliding on KQ, a pencil at P would trace a parabola whose focus is S and directrix XK. PROP. III. If a chord PQ (fig. 6) intersect the direc trix in Z, then SZ will be the external bisector of the angle PSQ. Join SP, SQ, and draw PM, QN per pendicular to the directrix. Then, because the triangles PMZ, QXZ are similar, PZ : QZ = Pit : QN = SP : SQ. .-. (Eucl. vi. A) SZ bisects the external angle of the triangle PSQ. Corollary. If the point Q move up to and become coincident with P, or if, in other words, the chord PQ become the tangent to the parabola at P, (hen the angle PSZ will become a right angle. PROP. IV. The tangent at any point of a parabola bisects the angle between the focal distance of the point and the perpendicular from the point on the directrix. Let PZ (fig. 7) be the tangent at P, meeting the directrix in Z ; then, if PM be drawn per pendicular to the directrix, it is easily seen that the two triangles SPZ, MPZ are equal in all respects, and the angle SPZ equal to the angle MPZ. If SM be joined, it can be shown that it is bisected at right angles by PZ, and that its middle point is the Fig. 7. point Y in Prop. ii. The line AY, it will be observed, is the tangent to the parabola at the vertex A. It appears, therefore, that the locus of the foot of the perpendi cular from the focus on the tangent at any point is the tangent at the vertex. It can also be seen that, if the tangent at P meet the axis in T, then SP = ST. For the angles STP, SPT are each equal to tho angle MPT, and therefore (Eucl. i. 6) SP, ST are equal. It may further be remarked that, if be any point in the tangent at P, then the triangles SPO, MPO are equal in all respects. If PN be drawn perpendicular to the axis to meet it ill N, then it will be seen that PN = 2AY and TN = 2AN = 2AT. Now, in the right-angled triangle TYS, TA : AY = AY : AS (Eucl. vi. 8), and therefore YA 2 = TA . AS Therefore PN 2 = 4YA 2 = 4TA . AS = 4AS.AN. If the normal PG be drawn meeting the axis in G, then the tri angles PNG, YAS are similar, and therefore NG-AS = PN:YA = 2:1 .-. NG = 2AS PROP. V. To draw a tangent to a parabola at a point on the curve. First Method. Take, a point T in the axis (fig. 7), such that ST is equal to SP, and join TP, Then STP will be the tangent at P. Second Method. Draw SZ at right angles to SP, meeting the directrix in Z. ZP is the tangent at P. Third Method. On SP as diameter describe a circle; this will touch the tangent at the vertex AY in a point Y. YP is tho tangent r,t P. PROP. VI. To draw a pair of tangents to a parabola from an external point. First Method. Let (fig. 8) be the point. Join OS, and with