Hence is derived an easy method of describing the ellipse by points.

From any point M in the major axis, draw an ordinate MQ to the circle, join CQ, cutting the circle described on Bb in q; through q draw qP parallel to AV, meeting MQ in P; then P will be a point in the ellipse.

as it ought to be. If Mp be taken equal MP, then will p also be a point in the ellipse. In like manner, two other points may be determined, equidistant from C to the left.

This process is not applicable to the hyperbola, since its equation can only be compared with the equation y = x2 — a2, which represents an equilateral hyperbola. The mode of constructing the latter must therefore be known, before we can employ it in the construction of the common hyperbola.

99. PROB. 5. To find the equations to the ellipse and hyperbola, when referred to a given system of conjugate diameters, the centre being the origin.

Let CP, CD, (fig. 49.) be any two semi-conjugate diameters, of which the former is supposed to be the axis of r, the latter that of y.

Assume CP d', CD=b'.

Then it may be proved precisely as in Art. 93, that the required equations are

a”y2±b”2x2 = ±a'b'2,

the upper sign being used for the ellipse, the lower for the hyperbola.

Observation. The equation to the hyperbola is deduced from that to the ellipse, by changing, as in Art. 95, the sign of b'2 When CP, CD are called conjugate diameters in the hyperbola, it is to be understood, that they are so called only by analogy, since CD does not, in fact, meet the curve. O

that is, if any ordinate Qv be drawn in the ellipse or hyperbola, and PC be produced to meet the curve in G,

Pv. vG Qv :: PC: CD2.

:

When the diameters PC, CD are rectangular, the above proportion becomes

AM. MV MP2 :: AC: BC.

:

The line 2p which has thus been assumed a third proportional to any two conjugate diameters, is called the parameter to the diameter PG. When the axes are rectangular, then 2p is called the principal parameter, or the latus rectum.

102. COR. 3. If y=mx be the equation to any diameter, then (86) the equation to the diameter conjugate to it will be either

according as the curve is referred to any system of conjugate diameters, or to the axes.

103. PROB. 6. To find the co-ordinates of the points, in which any diameter intersects the curve.

ON THE ELLIPSE AND HYPERBOLA.

Let y=mx be the equation to any diameter,

a22y2± b'2x2 = a22 b' that to the curve;

then, the co-ordinates required will be obtained by elimination between these two equations.

It appears, therefore, that in the ellipse, any diameter whatever will always meet the curve, and that in the hyperbola, a diameter can meet the curve only when b''-a'm' is positive,

or, m <

If m >

b

and if m =

the diameter meets the curve only at a point

a

infinitely distant.

Let Pp, Dd, be any two conjugate diameters, then, if Qq be drawn through P, parallel and equal to Dd, and C Q, Cq be joined, and produced to intersect the hyperbola at an those diameters which meet, the curve.

Z, z the diameters CZ, Cz will infinite distance, and will separate from those which can never meet,

The lines CZ, Cz are called asymptotes, and are defined by b'

the equation y = ± 7x.

104. COR. When the abscissa is supposed to be indefinitely great, it follows that the ordinate to the hyperbola must equal the ordinate to the asymptote.

105. Hence, if the equation to a curce be reducible to the form

y=ax+b+ + +

the straight line y = ax + b is an asymptote to the curve.

с d

For if x be assumed indefinitely great, then ++.. is

Ꮖ

indefinitely small, and, therefore, the ordinate to the curve becomes equal to the ordinate of the straight line y=ax+b.

106. It has already been remarked, that the equation to the ellipse becomes the equation to the hyperbola, when the sign of bor of b" in the former curve is changed. From the close analogy thus subsisting between these curves, it is evident, that the properties of the one must be deducible immediately from those of the other. For the sake of brevity, therefore, we shall, in what follows, comprehend both curves in the enunciation of any proposition, and shall deem it sufficient to give a demonstration in the case of the ellipse, which occurs the more frequently of the two in mathematical inquiries.

Any peculiarity that presents itself either in the nature or proof of the proposition, when applied to the hyperbola, shall be noted and explained.

ON THE PROPERTIES OF TANGENTS TO THE ELLIPSE AND HYPERBOLA.

107. DEF. A TANGENT to a line of the second order is a straight line that meets the curve, and being produced, does not cut it. That part of the axis, intercepted between the foot of the ordinate, and the point where the tangent meets the axis is called the sub-tangent.

Hence, if a straight line be drawn cutting the curve, and the two points of section be then supposed to coincide, the secant will become a tangent. It is on this principle that we shall resolve the following problem.

108. PROP. 1. To draw a tangent at a given point (x', y'), in the ellipse and hyperbola.

The equation to a secant drawn through this point, is

y-y=m(x-x') . . . . . . (1),

(m) being indeterminate.

......

..

Now, as r', y', are the co-ordinates of a point in the curve, they will satisfy the general equation

we have, therefore,

12

a22 y2 + b2x2 = ± a2b2;

12 12

a"2 yˆ± b2x" = ± a2ba‚