its form that there are two foci, S and H, and two directrices, MK, M'k, equally distant from the point C.

The point C is called the centre ; * A A', B B', the principal axes, or major and minor axes; and the points A, A', B, B', the vertices of the

curve.

Cor 1. If y' be any ordinate of the circle described on A A' as a diameter (radius a), corresponding to any ordinate y of the ellipse; then by (2) of the preceding, and Art. 30,

Whence it follows that the ellipse may be derived from the circle by diminishing proportionally, in the ratio a: b, all the ordinates relative to the same diameter. If the ellipse be compared with the inscribed circle, having for diameter the minor axis, it will be found in a similar way that the ordinate of the inscribed circle is to the corresponding ordinate of the ellipse as b: a. From this double comparison the construction of the ellipse by points, when the two axes are given, is easily effected by the aid of the two corresponding circles.

which represents a circle; hence, when its axes are equal, the ellipse becomes a circle.

Cor. 4. By Cor. 2, AS = a(1-e), and by the def., A S

= e AE,

that is, the sum of the focal distances of any point in the ellipse is constant, and equal to the major axis.

This property furnishes a simple method of determining any number of points in an ellipse of which the foci and principal axis are given. For if in A A' any point G be taken, and with centre S and radius A G

*The terms centre aud axes will be explained in a subsequent Article.

a circle be described, also with centre H and radius A'G another circle be described, then the points of intersection of these circles will evidently determine two points in the curve.

EXERCISES ON THE ELLIPSE.

1. Prove that the sum of two lines drawn from the foci to a point in the plane of the ellipse is greater or less than 2 a, according as the point is without or within the curve.

2. Prove that (fig. to Art. 40) AS. A'S

= b2.

3. Show that the square of any ordinate of a given ellipse varies as the rectangle of the corresponding segments of the major axis.

4. Deduce the equation of the ellipse from the property established in Cor. 4, and show that it is identical with that deduced in Art. 40.

5. Find the equation of the ellipse when the origin is placed at one of the foci.

THE HYPERBOLA.

41. To trace the curve represented by (1) Art. 37, when e is greater than unity.

e and 1 e2

=

Since 1 (1 − e) (1 + e), are both negative, the equation of the curve may be written in the form

Proceeding with this equation as with the analogous one of last Article,

is the equation of the curve when the middle point C of the line A A' is the origin of coordinates.

It

It may be shown, as in the preceding Articles, that the curve in this case is symmetrical with respect to the axes of x and y, and that it has two foci, S and H, and two directrices, as in the case of the ellipse. differs, however, from the ellipse in this respect, that the coefficients of x2 and y2 have opposite signs, which correspond geometrically to this,--that of the two right lines round which the curve is symmetrical, the one continues to meet the curve, but the other does not cut it; so that there exists only a single couple of vertices instead of two. And, moreover, the curve extends indefinitely to the right and left of the limits

VOL. I.

X

x = a, and x =

a, that is, it is composed of two parts with infinite branches, as in the figure.

This curve is called the hyperbola.*

The point C is the centre; A A', B B', the axes, or transverse and conjugate diameters; and A, A' the vertices.

The curve does not meet the axis of y, as has been stated, for when x = 0 in (2), y is imaginary. Hence B B' is sometimes called the

impossible axis. Also because

b2

a2

=

e-1, and as e may be of any

magnitude greater than unity, it is obvious that 6 may be either greater or less than a.

Cor. When b = a, the equation (2) becomes

The curve in this case is called the rectangular hyperbola, and it is to the ordinary hyperbola what the circle is to the ellipse.

EXERCISES ON THE HYPERBOLA.

=

2 a (e2 - 1).

1. Prove that the latus rectum of the hyperbola is 2. Prove that the difference of the focal distances of any point in the hyperbola is constant, and equal to 2 a.

3. Show that the square of any ordinate of the hyperbola varies as the rectangle of its distances from the extremities of the transverse axis.

42. To find the polar equation of the parabola, ellipse, and hyperbola.

Let S be the pole and Sx the prime radius (fig. to Art. 37). Put SP=r, angle PSx=0, and let the other lines be denoted as in Art. 37; then because SP = e. PM, and PM = EN = EA + AS + SN

2 m

1

which is the polar equation in general of the three curves.

In the parabola, since e = 1,

*Developing the equation (2) by the binomial theorem,

b

(1),

Now the larger becomes, the more y tends to become equal to - x, and when

= infinity, y = ± - x.

a

a

Hence if two lines be drawn through the origin, making angles whose tangents b are respectively and

a

b

with the axis of x, the curve will continually approxi

a

mate to these lines, but will meet them only at an infinite distance. These lines are called the asymptotes to the hyperbola.

43. To find the equation of the tangent to the parabola, ellipse, and hyperbola.

[The definitions of tangent and normal are the same as for the circle, Arts. 32, 33.]

Let (h, k) be a point P in the parabola

Proceeding with these equations as with (1) and (2) of Art. 32, there results for the tangent to (1) at the point (h, k), the equation

in which (h, k) is the point of contact, as has been stated, and (x, y) any point in the tangent PT.

a2 ky b2 h x

=

- a2 b2

(5),

is the analogous equation to the hyperbola ay b2 x2 = — a2 l2 (Art. 41).

Definition. The straight line intercepted between the ordinate PN and the point T, in which the tangent meets the axis of x, is called the subtangent.

Scholium.-As the normal PG by the definition is perpendicular to the tangent, and passes through the point of contact, its equation follows at once from that of the tangent and Arts. 16, 17.

The equations, therefore, of the normals of the preceding curves at the point (h, k) are, respectively,

Definition.-The line GN intercepted between the ordinate and

normal is called the subnormal.

h

=

Cor. 1. In (3), let y = 0, then x = AN; hence the subtangent in the parabola is bisected in the vertex. The negative sign merely implies that the intersection T is to the left of the origin A. Whence this simple method of drawing a tangent to a parabola from a given point P:

Take in N A produced, AT A N, the abscissa of the given point, and join TP, then is TP the required tangent.

Cor. 2. The equation of the perpendicular SY from the focus S, or (m, 0), on the tangent PT whose equation is (3), is, by Arts. 16, 17,

Eliminating y between (3) and (9), we get, since k2 = 4mh by the equation (1) of the curve, x=0. Hence a perpendicular from the focus of a parabola on any tangent intersects that tangent in the tangent from the vertex.

=

=

=

2 m.

Cor. 3. In (6), let y 2 m, or AG-AN Hence, in the parabola, the subnormal N G is equal to half the latus

.h 0; then x

AS+AT

=

SNNG= h.

rectum.

=

AS+AN = m + h, and SG

m + 2m = h+m; hence ST But by Art. 38 (fig. to Art. 37), SP = PM EN Whence if P be a point in the parabola, S the focus, and G, T the points of intersection of the normal and tangent at the point P with the axis of x, then S P = SG ST.

=

Cor. 5. Let a line Px' be drawn parallel to the axis; then because of the equals SP, ST by Cor. 4, and the parallels Px', Ax, the angle tPx' = STP SPT; and consequently since G P is perpendicular to the tangent PT, GPx' GPS. Hence the tangent and normal at any point of a parabola make equal angles with the focal distance of that point, and with a line drawn through it parallel to the axis.

In a similar way are all the properteis of the conic sections established.

EXERCISES ON TANGENTS AND NORMALS.

1. Prove that the tangents at the points B and B' in the ellipse (Art. 40) are perpendicular to the minor axis.

2. Let the ordinate NP (Art. 40) meet the circle described on A A' in Q, and let T be the intersection of the tangent to the ellipse at the point P, with A A' produced; then will the tangent to the circle at Q also pass through the point T.

3. Prove that the normal at any point of an ellipse bisects the angle contained by the focal distances of that point, and that the focal distances make equal angles with the tangent.

4. Show that in the hyperbola the normal at any point bisects the exterior angle between the focal distances of that point.