CHAPTER IV.

THE ELLIPSE.

THE ellipse has already been defined (p. 56) as the locus of a point which moves in a plane so that its distance from a fixed point in the plane is always in a constant ratio, less than unity, to its distance from a fixed line in the plane. The corresponding definition in the case of the parabola furnishes immediately the best condition for the geometrical construction of that curve, but this is not so with the ellipse. The ellipse can be more easily constructed geometrically from a property which will be shewn immediately to be involved in the above definition, and in virtue of which the curve may be defined as follows:

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DEF. The ellipse is the locus of a fixed point on a line of constant length moving so that its extremities are always on two fixed straight lines perpendicular to each other.

In Fig. 55 let ACA1, BCB1 be two straight lines intersecting each other at right angles in C. If a length (as ab) be marked off on the smooth edge of a slip of paper, and the slip be moved round so that the point a is always on the line BCB, and the point b on ACA,, then any point as P on the edge of the paper will trace out an ellipse. When the edge of the slip coincides with ACA, the tracing point will evidently be at a distance CA from C equal to aP, and when it coincides with BCB, the tracing point will be at a distance CB from C equal to bP. By this method of construction the curve is evidently symmetrical about both the lines ACA, and BCB1, i. e. if CA, be made equal to CA, A, will be a point on the curve, and if CB, be made equal

to CB, B, will be a point on the curve. It is moreover obvious that ACA, is the longest and BCB, the shortest line which can be drawn through C and terminated by the curve.

DEF. The line ACA, is called the major axis, the line BCB1 the minor axis, the point C the centre, and the points A, A, vertices of the curve.

From B, the extremity of the minor axis, as centre with radius CA (the semi-major axis), describe arcs cutting the major axis in F and F1; through B draw BM parallel to CA, from F draw FM perpendicular to BF meeting BM in M, and draw MX perpendicular to CA meeting it in X.

F will be the focus and MX the directrix (see definition, page 56).

From the similar triangles FBM, CFB,

CF FA CA: AX,

FA AX :: CF : CA :: FB : BM;

CF CA: CA: CX,

CA-CF CF+CA :: CX-CA: CA + CX,

or

i.e.

therefore A, B and A, are points satisfying the original definition.

DEF. A circle described on the major axis as diameter is called the auxiliary circle.

Through any point P on the ellipse draw the ordinate PN (perpendicular to major axis) meeting the axis in N and the auxiliary circle in Q. Since QN is parallel to BC and CQ = aP, aP is parallel to CQ,

..

but it is known that in the circle

QNAN. NA,,

.. PN2: AN.NA, :: BC|2 : AC2.

This is a very important property of the ellipse and will now be shewn to result from assuming the ratio FP NX to be constant.

:

Through P draw PA, PA, meeting the directrix in E and H. Join FH and draw PLK perpendicular to the directrix meeting FH in L and the directrix in K.

Since PK is parallel to à ̧X,

But by supposition

PL; PK :: FA, : A‚X

1

:: FA: AX.

FP PK :: FA : AX,

therefore FP = PL, and the angle LFP = FLP = the alternate angle LFX;

i.e.

FL bisects the angle PFX;

similarly FE bisects the angle between FX and PF produced, therefore the angle EFH is a right angle, since it is made up of the two angles EFX and HFX.

By the similar triangles PAN, A EX,

since

i. e,

EFH is a right angle;

PN is to AN, NA, in a constant ratio.

Hence taking PN coincident with BC, in which case

2

BC AC: FX : AX. A ̧X,

and .. PN2 : AN. NA, :: BC2 : AC2.

This of course shews that the point P is the same whether determined as the locus of a fixed point on a line of constant length

sliding between two fixed rectangular axes or as the locus of a point which moves so that its distance from a fixed point (F) is in a constant ratio to its distance from a fixed line (MX), i.e. the two definitions of the ellipse already given are really identical. From the symmetry of the curve it is evident that F1 is a second focus and MX, a second directrix.

1

Five geometrical conditions are generally necessary to determine an ellipse, and the ellipse shares with the hyperbola the property of satisfying five geometrical conditions. One or other of these curves can generally be drawn to pass through five given points or to touch five given straight lines, or to pass through two given points and touch three given lines, or to fulfil any five similar conditions. Which curve will satisfy the given conditions depends of course upon the relative positions of the given points and lines, and the necessary limitations will be noticed in discussing the separate problems. As in the case of the parabola the giving of certain points and lines is really equivalent in each case to the giving of two geometrical conditions; of these may be mentioned the centre, the foci, and the axes.

The eccentricity of the ellipse is (p. 57) the numerical value of the above fixed ratio; it is generally denoted by e and calling

as is evident from the similar triangles FBM, FCB.

PROBLEM 60. (Fig. 55.) To describe an ellipse having given axes AA,, BB,.

First Method. Draw two lines perpendicular to each other intersecting in C. Set off CA, CA, each equal to 1⁄2 AA,, and CB, CB, each equal to BB,. Take a smooth edged slip of paper and mark off on it Pa = CA and Pb=CB (a and b may be on the same or on opposite sides of P). Keep the point a on the minor axis and the point b on the major axis and (as already demon

strated) the point P will be on the curve. Any number of points may thus be determined. In the lower portion of the figure the

lengths CA, CB are shewn set off on opposite sides of P, and this arrangement is the better when the lengths AA,, BB, are nearly equal, as in that case, when set off on the same side of P, the distance ab is too short to determine the direction of Pa with accuracy.

Second Method (fig. 56). Arrange the axes as above, and on each as diameter describe a circle. Draw any number of radii as C1, C2, &c. From the extremities of the radii of the circle on the major axis draw lines parallel to the minor axis, and from the ends of the radii of the circle on the minor axis draw lines parallel to the major axis. The lines drawn from corresponding points (as 7P, 7'P) will intersect on the required ellipse, which can therefore be drawn through the points thus determined.

The proof is at once obvious by drawing through any point P on the curve a line parallel to the corresponding radius C7, cutting the axes in b and a. Then

Ca P7 is a parallelogram, and .. Pa = C7 = CA,

Cb P7' is a parallelogram, and .. Pb = C7' CB,