An ellipse may also be defined as the locus of a point moving so that the sum of its distances from two fixed points is constant.

Exercises.-1. Assume a point and a straight line and construct PF 3 by points an ellipse having the ratio PM

2. Two points are 10 centimetres apart. Construct by continuous motion an ellipse the major axis of which is 12 centimetres.

3. From any point within an ellipse the sum of the distances to the foci is less than the major axis, and from any point without an ellipse the sum of the distances to the foci is greater than the major axis.

P

F

FIG. 359.

NOTE. — There are so many ways of constructing an ellipse with reasonable accuracy that there is no excuse for making up a combination of arcs of circles and calling the figure an ellipse.

167. THEOREM. If a straight line be drawn through a point of an ellipse so as to make equal angles with the focal radii to the same point, it will be a tangent to the curve. A tangent is the limiting position toward which a secant approaches as the two points of intersection approach coincidence.

When the secant shall

have become a tangent,

all of its points except the 4

point of contact will lie

without the curve.

R

H

1Q

F

FIG. 360.

Produce F'P so that PH shall equal PF; making PHF an isosceles triangle. Through P draw RQ1 to FH; it will make equal angles with PF and PF'.

If RQ be a tangent, every point except the point of contact P must be shown to be exterior to the ellipse. Let R represent any point other than P.

But

F'R+RH>F'H.

RH RF and F'H= AA'.

.. F'R+RF > AA'.

(§ 167, Ex. 3)

Hence the point R, which may be any point on the line other than P, is exterior to the curve, and RQ is a tangent.

NOTE. — If an ellipse be revolved about its major axis, a prolate ellipsoid of revolution is generated. If revolved about its minor axis, an oblate ellipsoid of revolution is formed. The earth is approximately an oblate ellipsoid of revolution.

168. THEOREM I. An ellipse is determinable when its axes are given.

We have seen that an ellipse is the locus of a point which moves so that the sum

of its distances from F and F' equals AA' (2 a).

When the moving point is

F

B

FIG. 361.

A

F

If, then, the segments which are to be axes are placed so as to mutually bisect each other at right angles, the foci may be determined by constructing an arc of a circle with B as centre and a as radius. The foci and the major axis determine the ellipse (§ 166); therefore the axes do.

THEOREM II. A section oblique to the elements of a right circular cylinder will be an ellipse.

Let APA' represent the oblique section. Within the cylinder and tangent to the secant plane, conceive spheres to be inscribed as indicated in the figure; they will be tangent to the secant plane at two points, as F and F".

COROLLARY. Any right section is a circle; so that an ellipse may be projected into a circle.

169. PROBLEM. If a and b represent the semi-axes of an ellipse, show that the area is represented by ab.

Let APA' represent an ob

π

A

lique section of a right circular

Β'

cylinder and KHK' a right section of the same passed through K' the middle point of AA'.

BB' will be perpendicular to A AA' and will be the minor axis of the ellipse, as well

as being the diameter of the

KHK'.

FIG. 363.

K

H

B

From P draw PQL to CB and PH perpendicular to

the plane of the circle.

QH will also be perpendicular to CB (§ 112).

the

If the point P should move from B to A to A', segment PQ remaining perpendicular to BB' would generate half the area of the ellipse; and the segment HQ would generate half the area of the circle.

Each line segment would remain perpendicular to BB', would move the same distance, and the ratio of their lengths would remain the same, therefore

Exercises. - 1. Deduce the same expression for the area of an ellipse by inscribing a polygon within the ellipse, comparing its area with the area of the projected polygon in the circle, and then by the theory of limits determine the relation between the areas of the ellipse and the circle.

2. Show that an ellipse has a centre, i.e. a point through which, if chords be drawn, they will be bisected.

3. Show that if a line be drawn from the centre of the ellipse to the point Q, in the figure of § 167, its length will be a. Hence the locus of the foot of a perpendicular from a focus to a tangent is the circumference of a circle on the major axis as a diameter.

This circle is called the director circle.

NOTE. The earth's meridians are ellipses, the axis of the earth being the minor axis of each ellipse.

It is this fact which gives rise to the statement that the Mississippi River runs up hill. Its mouth is further from the centre of the earth than its source.

The locus of the earth in its annual motion about the sun is approximately an ellipse with the sun at one focus.

If a light (as an electric arc-light) were placed at one focus of a prolate ellipsoid, all rays reflected from the surface would meet at the other focus. (Established by § 167.)

Whispering galleries are also made which depend upon this principle.

PARTICULAR CASES.

170. If a plane which intersects the surface of a cone in an ellipse be moved parallel to itself until it passes through the vertex, the ellipse will degenerate to a point. If the plane be moved further, an ellipse in the other nappe will be the result.

If a plane which cuts an ellipse from a right circular cone be rotated about some axis toward the position of being perpendicular to the axis, the foci will approach each other, and when the plane becomes perpendicular to the axis, the foci and centre will coincide and the ellipse will become a circle.

THE HYPERBOLA.

171. Definitions. If a plane intersect a right circular cone and make with the plane of a circular section an angle greater than that made by the elements of the cone with the same section, it will intersect both nappes of the cone.

The line of intersection is called a hyperbola. In general it is in two branches, neither of which close.