Hence the portion of the parabola included between two ordinates indefinitely near is double of the corresponding portion of the external space ABV. The same may be proved for every point of the curve, and hence the whole space AVG is double the space ABV. Whence AVG is two thirds of ABVG, and the parabolic segment AVD is two thirds of the circumscribing rectangle ABCD. Therefore a segment, etc.

EXERCISES ON THE PARABOLA.

1. The diameter of the circle described about the triangle AVB is equal to 5FV. (See fig., Pr. 4.)

2. If from the point D, DE be drawn at right angles to FA, then AE is equal to 2VF. (See fig., Pr. 7.)

3. If the triangle ADF is equilateral, then AF is equal to the latus rectum. (See fig., Pr. 7.)

4. If AB is a common tangent to a parabola, and the circle described on the latus rectum as a diameter, prove that AF and BF make equal angles with the latus rectum,

5. If the tangent AC meets the directrix in G, prove that AC. AG-AF2, and that AC.CG-AF.FV. (See fig., Pr. 3.)

6. If AE be drawn at right angles to AV, meeting the axis in E, then CE is equal to 4VF. (See fig., Pr. 7.)

+ 7. The tangent at any point of a parabola meets the directrix and latus rectum produced in points equally distant from the focus.

8. Prove that BC=CD, and that BA.BC=BF.BD. (See fig., Pr. 8.)

9. If a circle be described about the triangle AFC, the tangent to it from V is equal to one half AC. (See fig., Pr. 7.)

10. If the ordinate of a point A bisect the subnormal of a point B, the ordinate of A is equal to the normal of B.

11. If from any point on the tangent to a parabola a line be drawn touching the parabola, the angle between this line and the line to the focus from the same point is constant.

12. If the diameter AC meets the directrix in G, and the chord drawn through the focus parallel to the tangent at A in C, prove that AC-AG. (See fig., Pr. 14.)

13. Required the area of a segment of a parabola cut off by a chord 15 inches in length, perpendicular to the axis, the corresponding abscissa of the axis being 21 inches.

14. An ordinate to the axis of a parabola is 9 inches, and the corresponding abscissa is 10 inches; required the latus rectum. K

15. An ordinate to a diameter of a parabola is 12 inches, and the corresponding abscissa is 5 inches; required the parameter of that diameter.

16. The latus rectum of a parabola is 20 inches; required the area of the segment cut off by a double ordinate to the axis when the corresponding abscissa is 30 inches.

17. The latus rectum of a parabola is 9. to the axis corresponding to the abscissa 4?

What is the ordinate

18. The latus rectum of a parabola is 10 inches. Find the ordinate to the axis corresponding to that point of the curve from which, if a tangent and normal be drawn, they will form with the axis a triangle whose area is 36 inches.

19. The latus rectum of a parabola is 15, and a tangent is drawn through the point whose ordinate to the axis is 4. Determine where the tangent line meets the axis produced.

20. The latus rectum of a parabola is 12, and a tangent is drawn through the point whose ordinate to the axis is 7. Determine where the normal line passing through the same point meets the axis.

ELLIPSE.

Definitions.

1. An ellipse is a plane curve traced out by a point which moves in such a manner that the sum of its distances from two fixed points is always the same.

D

2. The two fixed points are called the foci of the ellipse. Thus, if F and F' are two fixed points, and if the point D moves about F in such a manner that the sum of its distances from F and F' is always the same, the point D will describe an ellipse, of which F and F' are the foci.

3. The centre of the ellipse is the mid

dle point of the straight line joining the foci.

F'

F

4. The eccentricity is the distance from either focus to the

centre.

centre of the ellipse, and CF or CF' is the eccentricity.

5. A diameter is any straight line passing through the centre, and terminated on both sides by the curve.

Δ'

6. The extremities of a diameter are called its vertices. Thus, through C draw any straight line DD' terminated by the curve; DD' is a diameter of the ellipse; D and D' are the vertices of that diameter.

7. The major axis is the diameter which passes through the foci.

8. The minor axis is the diameter which is perpendicular to the major axis.

Thus, produce the line FF" to meet the curve in A and A', and through C draw BB' perpendicular to AA'; then is AA' the major axis, and BB' the minor axis.

9. A tangent to an ellipse is a straight line which meets the curve in one point only, and every where else falls without it.

10. An ordinate to a diameter is a straight line drawn from any point of the curve to the diameter, and is parallel to the tangent at one of its vertices.

D

T

G
E

Thus, let DD' be any diameter, and TT' a tangent to the ellipse at D. From any point G of the curve draw GKG' T parallel to TT', and cutting DD'in K; then is GK an ordinate to the diameter DD'. It is proved in Pr. 7 that the tan gents at D and D' are parallel.

It is proved in Pr. 21, Cor. 1, that GK is equal to G'K; hence the entire line GG' is called a double ordinate.

11. Each of the parts into which a diameter is divided by an ordinate is called an abscissa.

Thus, DK and D'K are the abscissas of the diameter DD' corresponding to the ordinate GK, or to the point G.

12. One diameter is said to be conjugate to another when it is parallel to the ordinates of the other diameter.

Thus, draw the diameter EE' parallel to GK, an ordinate to the diameter DD', in which case it will, of course, be parallel to the tangent TT'; then is the diameter EE' conjugate to DD'.

I

T

F

G

T

13. The latus rectum is the double ordinate to the major axis which passes through one of the foci.

Thus, through the focus F' draw LL', a double ordinate to the major axis; it will be the latus rectum of the ellipse.

14. A subtangent is that part of an axis produced which is included between a tangent and the ordinate drawn from the point of contact.

Thus, if TT' be a tangent to the curve at D, and DG an ordinate to the major axis, then GT is the corresponding subtangent. 15. The directrix of an ellipse is

through one extremity of the latus rectum LL', meeting the axis produced in T, and GT be drawn through the point of intersection perpendicular to the axis, it will be the directrix of the ellipse. The ellipse has two directrices, one corresponding to the focus F, and the other to the focus F'.

PROPOSITION I. THEOREM.

The sum of the two lines drawn from any point of an ellipse to the foci is equal to the major axis.

Let ADA' be an ellipse, of which F, F' are the foci, AA' is the major axis, and D any point of the curve; then will DF+DF' be equal to AA'.

For, by Def. 1, the sum of the distances of any point of the curve from the foci is equal to a given line. Now,

A

D

A

F'

C

F

when the point D arrives at A, FA+F'A, or 2AF+FF" is equal to the given line.

And when D is at A', FA'+F'A', or 2A'F'+

FF' is equal to the same line. Hence

consequently,

2AF+FF'=2A'F'+FF';

AF is equal to A'F'.

Hence DF+DF', which is equal to AF÷AF', must be equal to AA'. Therefore the sum of the two lines, etc.

Cor. The major axis is bisected in the centre. For, by Def. 3, CF is equal to CF'; and we have just proved that AF is equal to A'F'; therefore AC is equal to A'C.

The major axis and foci of an ellipse being given, to describe the curve.

FIRST METHOD. By points.

Let AA' be the major axis, and F, F' the foci of an ellipse. Take E any point between the foci, and from F and F' as centres, with the distances AE, A'E A as radii, describe two circles cutting each other in the point D; D will be a point on the ellipse. For, join FD,

F'D; then DF+DF'=EA+EA'=AA'; and, at whatever point between the foci E is taken, the sum of DF and DF' will be equal to AA'. Hence, by Def. 1, D is a point on the curve; and, in the