PROPOSITION XXIII. THEOREM.

The area of an ellipse is a mean proportional between the two circles described on its axes.

Let AA' be the major axis of an ellipse ABA'B'. On AA' as a diameter describe a circle; inscribe in the circle any regular polygon AEDA', and from the vertices E, D, etc., of the polygon draw perpendiculars to AA'. Join the points B, G, etc., in which these perpendiculars intersect the ellipse, and there will be inscribed in the ellipse a polygon of an equal number of sides. Now the area of the trapezoid CEDH is equal to (CE+DH) x

CH

2

X

; and the area of the trapezoid CBGH is equal to (CB+GH)

CH. These trapezoids are to each

2

other as CE+DH to CB+GH, or as AC to BC (Pr. 14).

In the same manner, it may be proved that each of the trapezoids composing the polygon inscribed in the circle is to the corresponding trapezoid of the polygon inscribed in the ellipse as AC to BC. Hence the entire polygon in

D

G

H

B'

scribed in the circle is to the polygon inscribed in the ellipse as AC to BC.

Since this proportion is true, whatever be the number of sides of the polygons, it will be true when the number is indefinitely increased; in which case one of the polygons coincides with the circle, and the other with the ellipse. Hence we have

area of circle: area of ellipse:: AC: BC.

But the area of the circle is represented by AC2; hence the area of the ellipse is equal to TAC× BC, which is a mean proportional between the two circles described on the axes.

PROPOSITION XXIV. THEOREM.

The distance of any point in an ellipse from either focus is to its distance from the corresponding directrix as the eccentricity to half the major axis.

Let D be any point in the ellipse; let DF, DF' be drawn to the two foci, and DG, DG' perpendicular to the directrices; then DF: DG:: DF': DG':: CF: CA.

tween FE and F'E. By B. IV., Pr. 34,

FF': DF'+DF:: DF'-DF: F'E-FE.

Dividing each term by two, we have

In the same manner, it may be proved that

DF': DG':: CF: CA.

EXERCISES ON THE ELLIPSE.

1. If a series of ellipses be described having the same major axis, the tangents at the extremities of their latera recta will all meet the minor axis in the same point.

2. The foci of an ellipse being given, it is required to describe an ellipse touching a given straight line.

3. If the angle FBF' be a right angle, prove that CA2=2CB2. (See fig., Pr. 5.)

4. If a circle be described touching the major axis in one focus, and passing through one extremity of the minor axis, AC will be a mean proportional between BC and the diameter of this circle. (See fig., Pr. 5.)

5. If, on the two axes of an ellipse as diameters, circles be described, and a line be drawn through the centre cutting the larger circle in H and H', and the smaller circle in K and K', then HK.H'K=CF2. (See fig., Pr. 14.)

6. If DG produced meet the tangent at the extremity of the latus rectum in K, then KG-DF. (See fig., Pr. 11.)

7. A tangent to the ellipse makes a greater angle with a line drawn from the point of contact to one of the foci than with the perpendicular on the directrix. (See fig., Pr. 24.)

8. If from C one line be drawn parallel, and another perpendic ular to the tangent at D, they inclose a part of DF' equal to DF. (See fig., Pr. 9.)

9. If the tangent at the vertex A cut any two conjugate diameters in T and t, then AT.At-BC2. (See fig., Pr. 16.)

10. What is the area of an ellipse whose axes are 46 and 34 feet?

11. An ordinate to the major axis of an ellipse is 7 inches, and the corresponding abscissas are 5 and 20 inches; required the la tus rectum.

12. The latus rectum of an ellipse is 11 inches, and the major axis 26 inches; required the area of the ellipse.

13. The eccentricity of an ellipse is 10 inches, and its latus rectum 12 inches; required the area of the ellipse.

14. Supposing a meridional section of the earth to be an ellipse whose major axis is 7926 miles, and its minor axis 7900 miles, what is the area of the section?

15. What is the latus rectum of the terrestrial ellipse, and what is its eccentricity?

16. What is the distance of the directrix of the terrestrial ellipse from the nearest vertex of the major axis?

17. If the axes of an ellipse are 60 and 100 feet, what is the radius of a circle described to touch the curve, when its centre is in the major axis at the distance of 16 feet from the centre of the ellipse? Ans. 27.495 feet.

18. If the axes of an ellipse are 60 and 80 feet, what are the areas of the two segments into which it is divided by a line perpendicular to the major axis at the distance of 10 feet from the centre? Ans. 1291.27 and 2478.65 feet.

19. The minor axis of an ellipse is 8 inches, the latus rectum 5 inches, and an ordinate of 3 inches is drawn to the major axis; determine where the tangent line drawn through the extremity of this ordinate meets the major axis produced.

20. Determine where the tangent line in the last example meets the minor axis produced.

HYPERBOLA.
Definitions.

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

2. The two fixed points are called the foci of the hyperbola.

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

If the point D' moves about F'in such a manner that D'F-D'F' is always equal to DF'-DF, the point D' will describe a second branch of the curve similar to the first. The two branches are called branches of the hyperbola.

3. The centre of the hyperbola is the middle point of the straight line joining the foci.

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

B

F

A CA

D

B'

-F

Thus, let F and F' be the foci of an hyperbola. Draw the line FF', and bisect it in C. The point C is the centre of the hyperbola, and CF or CF' is the eccentricity.

5. A diameter is any straight line passing through the centre, and terminated on both sides by opposite branches of an hyperbola.

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

7. The transverse axis is the diameter which, when produced, passes through the foci.

8. The conjugate axis is a line drawn through the centre perpendicular to the transverse axis, and terminated by the circum

ference described from one of the vertices of the transverse axis as a centre, and with a radius equal to the eccentricity.

Thus, through C draw BB' perpendicular to AA', and with A as a centre, and with CF as a radius, describe a circumference cutting this perpendicular in B and B'; then AA' is the transverse axis, and BB' the conjugate axis.

If, on BB' as a transverse axis, opposite branches of an hyperbola are described, having AA' as their conjugate" axis, this hyperbola is said to be conjugate to the former.

9. A tangent to an hyperbola 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 meet the diameter produced, and is parallel to the tangent at one of its vertices.

Thus, let DD' be any diameter, and TT' a tangent to the hyperbola at D. From any point G of the curve draw GKG' parallel to TT', and cutting DD' produced in K; then is GK an ordinate to the diameter DD'.

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.

FA

B

K

F

T

E' B

11. The parts of the diameter produced, intercepted between its vertices and an ordinate, are called its abscissas.

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

12. When the ordinates of a diameter of an hyperbola are parallel to a diameter of the conjugate hyperbola, the latter diameter is said to be conjugate to the former.

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'

T