PARABOLA, in geometry, a figure arising from the section of a cone, when cut by a plane parallel to one of its sides.
See CONIC SECTIONS.
To describe a parabola in piano, draw a right line A B (Plate Parabola, fig. 1) and assume a point C without it ; then, in the same plane with this line and point, place a square rule D E F, so that the side D E may be applied to the right line A B, and the other E F turned to the side on which the point C is situated. This done, and the thread F G C, exactly of the length of the side of the rule, E F, being fixed at one end to the extremity of the rule F, and at the other to the point C, if you slide the side of the rule, D E, along the right line A B, and by means of a pin, G, continually apply the thread to the side of the rule, E F, so as to keep it always stretched as the rule is moved along, the point of this pin will describe a parabola G H 0.
Definitions. 1. The right line A B is called the directrix. 2. The point C is the focus of the parabola. 3. All per pendiculars to the directrix, as L K, M 0, &c. are called diameters ; the points, where these cut the parabola, are called its vertices ; the diameter B I, which pas ses through the focus C, is called the axis of the parabola ; and its vertex, H, the principal vertex. 4. A right line, terminated on each side by the parabola, and bisected by a diameter, is called the ordinate applicate, or simply the ordinate, to that diameter. 5. A line equal to four times the segment of any diameter, in tercepted between the directrix and the vertex where it cuts the parabola, is call ed the latus rectum, or parameter of that diameter. 6. A right line which touches the parabola only in one point, and being produced on each side falls without it, is a tangent to it in that point.
Prop. 1. Any right line, as G E, drawn from any point of the parabola, G, per pendicular to A B, is equal to a line, G C, drawn from the same point to the focus. This is evident from the description ; for the length of the thread, F G C, being equal to the side of the rule El F, if the part F G, common to both, be taken away, there remains E G = G C. Q. E. D.
The reverse of this proposition is equally evident, viz. that if the distance of any point from the focus of a para. bola be equal to the perpendicular drawn from it to the directrix, then shall that point fall in the curve of the parabola.
Prop. 2. If from a point of the parabola, D, (fig. 2) a right line be drawn to the focus, C ; and another D A, perpendicular to the directrix ; then shall the right line, D E, which bisects the angle, A D C, con tained between them, be a tangent to the parabola in the point D: a line also, as H K, drawn through the vertex of the axis, and perpendicular to it, is a tangent to the parabola in that point.
1. Let any point F, be taken in the line D E, and let F A, F C, and A C be joined ; also let F G be drawn perpendicular to the directrix. Then, because (by Prop. 1), D A= D C, D F common to both, and the angle F I) A= F D C, F C will be equal to F A ; but F A greater than F G, therefore F C greater than F G, and consequently the point, F, falls without the parabola : and as the same can be de monstrated of every other point of D E, except D, it follows that D E is a tangent to the parabola in D. Q. E. D.
2. If every point of H K, except H, falls without the parabola, then is II K a tangent in H. To demonstrate this, from any point K, draw K L perpendicular to A B, and join K C ; then because K C is greater than CH= HB K L, it fol lows that K C is greater than K L, and consequently that the point K falls with out the parabola: and as this holds of every other point, except H, it follows that K H is a tangent to the parabola in H. Q. E. D.
Prop. 3. Every right line, parallel to a tangent, and terminated on each side by the parabola, is bisected by the diameter passing through the point of contact: that is, it will be an ordinate to that cliame• ter. For let E e (fig. 3 and 4) termina ting in the parabola in the points E e, be parallel to the tangent I) K ; and let A D be a diameter passing through the print of contact D, and meeting E e in L ; then Let A D meet the directrix in A, and from the points E e, let perpendiculars E F, e f, be dittwn to the directrix ; let C A be drawn, meeting B e in C; and on the centre E, with the distance E C, let a circle be described, meeting A C again in H, and touching the directrix in F ; and let D C be joined. Then because D A = I) C, and the angle A D K = the angle C D K, it follows (4. 1) that D K per pendicular to A C; wherefore E e perpen dicular to A C, and C G = G H (3. 3) ; so that eC = e H (4. 1), and a circle des cribed upon the centre e with the radius e C, must pass through H; and because e C = e f, it must likewise pass through f Now because F f is a tangent to both these circles, and A H C cuts them, the square A F= the rectangle C A H (36. 3) = the square A f;- therefore A F = At, and F E, A L, and f e are parallel ; and consequently L E = L e. Q E. D.