Elements of Conic Sections |
Other editions - View all
Common terms and phrases
a²y² AC² algebraic expression angle FMI APXA'P asymptotes axes b²x² BC² bisects the angle CA² centre circle CM² CN² CNXCY conic surface conjugate axis consequently corresponding abscissas denoted diameter directrix double ordinate drawn perpendicular ellipse ellipse ABA'B equation extremities F and F F'M+FM find the algebraic foci focus Geom given line hence Geom hyperbola intersecting M'KO M'O² M'OK meet the asymptotes MF'P MP² ordinate be drawn ordinate MP P'ARK parabola parallel parallelogram parameter PARM perpendicular to EG point of contact points F polygon produced to meet Prop proportion PROPOSITION radii radius of curvature radius vector right angles semi-conjugate axis similar triangles straight line drawn subnormal subtangent tangent TT THEOREM transverse axis vertex x²-a² сх
Popular passages
Page 84 - THEOREM The area of an ellipse is equal to the product of its semi-axes multiplied by the circumference of a circle whose diameter is unity.
Page 65 - AA' ; then is AA' the major axis, ar.d BB' the minor axis. 9. A tangent is a stiaight line which meets the curve, but, Oeing produced, does not cut it. 10. An ordinate to a diameter, is a straight line drawn from any point of the curve to the diameter, parallel to the tangent at one of its vertices. Thus, let DD' be any diameter, and TT' a tangent to the ellipse at D.
Page 21 - N3 6. A straight line which meets the curve in any point, but which, when produced both ways, does not cut it, is called a tangent to the curve at that point. 7. A straight line drawn from any point in the curve, parallel to the tangent at the vertex of any diameter, and terminated both ways by the curve, is called an ordinate to that diameter.
Page 1 - MF' be drawn, their sum MF+MF' is equal to a given line. II. The straight line drawn through the foci, and terminated by the curve, is called the transverse or major axis. The middle of that part of the transverse axis which lies between the foci, is called the centre of the ellipse. The straight line drawn through the centre, at right angles to the transverse axis, and terminated by the curve, is called the conjugate or minor axis. Thus, if the straight line joining F and F...
Page 6 - A2 hence the vertices of the transverse axis, hence the square of any ordinate is to the product of its distances from the vertices of the transverse axis as the square of the conjugate axis is to the square of the transverse...
Page 39 - ... of the two angles ABD, ADB, or to the sum of the two angles BAF, ADB. Take away the common angle BAF, and we have the angle DAF equal to ADF. Hence the line AF is equal to FD. Therefore, if a circle be described with the center F, and radius FA, it will pass through the three points B, A, D. Cor. 2. The normal bisects the angle made by the diameter at the point of contact, with the line drawn from that point to the focus. For, because BD is parallel to CE, the alternate angles ADF, DAE are equal....
Page 33 - The straight line drawn through the for.us, perpendicular to the directrix, is called the axis. The point in which the axis intersects the curve, is called the vertex of the parabola. Thus, BX, drawn through F, perpendicular to EG, is the axis; and A, the middle (Def. I.) of the perpendicular FB, is the vertex of the parabola MAM".
Page 4 - Take a thread longer than the distance FF', and fasten one of its extremities at F, the other at F'. Then let a pencil be made to glide along the thread so as to keep it always stretched ; the curve described by the point of the pencil will be an ellipse. For, in every position of the pencil, the sum of the distances DF, DF' will be the same, viz., equal to the entire length of the string.
Page 47 - AN hyperbola is a plane curve, in which the difference of the distances of each point from two fixed points, is equal to a given line. 2. The two fixed points are called the foci. Thus, if F and F...
Page 8 - As the Square of the Transverse Axis : Is to the Square of the Conjugate : : So is the Rectangle of the Abscisses : To the Square of their Ordinate.