HYPERBOLA.

Definitions.

1. AN hyperbola is a plane curve, in which the difference of the distances of each point from two fixed points, is equal to a given line.

2. The two fixed points are called the foci. 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

F

always equal to DF-DF, the point D' will describe a second hyperbola similar to the first. The two curves are called opposite hyperbolas.

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

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

Thus, let F and F be the foci of two opposite hyperbolas. Draw the line FF, and bisect it in C. The point C is the center of the hyperbola, and CF or CF is the eccentricity.

5. A diameter is a straight line drawn through the center, and terminated by two opposite hyperbolas.

6. The extremities of a diameter are called its vertices.

B

F

-F

A

CA

D'

B'

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

7. The major axis is the diameter which, when produced, passes through the foci; and its extremities are called the principal vertices.

8. The minor axis is a line drawn through the center per

pendicular to the major axis, and terminated by the circumference described from one of the principal vertices as a center, and a radius equal to the eccentricity.

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

If on BB' as a major axis, opposite hyperbolas are described, having AA' as their minor axis, these hyperbolas are said to be conjugate to the former.

9. A tangent is a straight line which meets the curve, but, being produced, does not cut it.

10. An ordinate to a diameter, is a straight line drawn from any point of the curve to meet the diameter produced, 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 FA is GK an ordinate to the diameter DD'.

It is proved, in Prop. XIX., Cor. 1, that GK is equal to

D'

B

F

T'

FB

G/K; hence the entire line GG' is called a double ordinate. 11. The parts of the diameter produced, intercepted be tween 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. Two diameters are conjugate to one another, when each is parallel to the ordinates of the other.

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'.

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 hyperbola.

T

G

T

15. A subtangent is that part of the 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.

To describe an hyperbola.

Let F and F be any two fixed points. Take a ruler longer than the distance FF', and fasten one of its extremities at the point F'. Take a thread shorter than the ruler, and fasten one end of it at F, and the other to the end H of the ruler. Then move the ruler HDF/

HAN

F

about the point F', while the thread is kept constantly stretched by a pencil pressed against the ruler; the curve described by the point of the pencil, will be a portion of an hyperbola. For, in every position of the ruler, the difference of the lines DF, DF will be the same, viz., the difference between the length of the ruler and the length of the string.

If the ruler be turned, and move on the other side of the point F, the other part of the same hyperbola may be described. Also, if one end of the ruler be fixed in F, and that of the thread in F', the opposite hyperbola may be described.

PROPOSITION II. THEOREM.

The difference of the two lines drawn from any point of an hyperbola to the foci, is equal to the major axis.

Let F and F be the foci of two opposite hyperbolas, AA' the major axis, and D any point of the curve; then will DF-DF be equal to AA'.

For, by Def. 1, the difference of the distances of any point of the curve from the foci, is equal to a given line. Now when the point D arrives at A,

F

A CAF

F/A-FA, or AA/+F'A'-FA, is equal to the given line. And when D is at A', FA-F'A', or AA/+AF-A/F', is equal to the same line. Hence

or

AA+AF-A'FAA'+F'A'-FA,
2AF=2A/F';

that is, AF is equal to A/F'.

Hence DF-DF, which is equal to AF-AF, must be equal to AA'. Therefore, the difference of the two lines, &c. Cor. The major axis is bisected in the center. 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.

PROPOSITION III. THEOREM.

Every diameter is bisected in the center.

Let D be any point of an hyperbola; join DF, DF', and FF'. Complete the parallelogram DFD'F', and join DD'.

F

Now, because the opposite sides of a parallelogram are equal, the difference between DF and DF is equal to the difference between D/F and D'F'; hence D' is a point in the opposite hyperbola. But the diagonals of a parallelogram bisect each other; therefore FF is bisected in C; that is, C is the center of the hyperbola, and DD' is a diameter bisected in C. Therefore, every diameter, &c.

PROPOSITION IV. THEOREM.

Half the minor axis is a mean proportional between the distances from either focus to the principal vertices.

Let F and F be the foci of opposite hyperbolas, AA' the major axis, and BB' the minor axis; then will BC be a mean proportional between AF and A'F.

Join AB. Now BC is equal to AB'AC, which is equal to FC-AC (Def. 8). Hence (Prop. X., B. IV.),

and hence

B

F

-F

CA

B'

BC2=(FC-AC) × (FC+AC)
=AFXA/F;

AF: BC:: BC: A/F.

Cor. The square of the eccentricity is equal to the sum of the squares of the semi-axes.

For FC is equal to AB2 (Def. 8), which is equal to AC'+ BC'.

PROPOSITION V. THEOREM.

A tangent to the hyperbola bisects the angle contained by lines drawn from the point of contact to the foci.

Let F, F be the foci of two opposite hyperbolas, and D any point of the curve; if through the point D, the line TT' be drawn bisecting the angle FDF'; then will TT be a tangent to the hyperbola at D.

For if TT' be not a tangent, let it meet the curve in some other point, as E. Take DG equal to DF; and join EF, EF', EG, and FG.

F

Now, in the two triangles DFH, DGH, because DF is equal to DG, DH is common to both triangles, and the angle FDH is, by supposition, equal to GDH; therefore HF is equal to HG, and the angle DHF is equal to the angle DHG. Hence the line TT is perpendicular to FG at its middle point; and, therefore, EF is equal to EG.

Now F/G is equal to F/D-DF, or F/E-EF, from the nature of the hyperbola. But F/E--EG is less than F'G (Prop. VIII., B. I.); it is, therefore, less than F'E-EF. Consequently, EG is greater than EF, which is impossible, for we have just proved EG equal to EF. Therefore E is not a point of the curve; and TT' can not meet the curve in any other point than D; hence it is a tangent to the curve at the point D. Therefore, a tangent to the hyperbola, &c.

Cor. 1. The tangents at the vertices of the axes, are per pendicular to the axes; and hence an ordinate to either axis is perpendicular to that axis.

Cor. 2. If TT represent a plane mirror, a ray of light proceeding from F in the direction FD, would be reflected in a line which, if produced, would pass through F', making the angle of reflection equal to the angle of incidence. And, since the hyperbola may be regarded as coinciding with a tangent at the point of contact, if rays of light proceed from one focus of a concave hyperbolic mirror, they will be reflected in lines diverging from the other focus. For this reason, the points F, F' are called the foci.