COR. 2. Let the point be within the curve, and suppose Nn to be a diameter bisecting the chord Mm; we have, then, by the last Cor.

which is no other than the equation to the ellipse and hyperbola, when referred to conjugate diameters (100).

The general equation to the parabola (164) may, in like manner, be shewn to be a particular case of the proposition under consideration.

COR. 3. Let the secants move parallel to themselves, until they become tangents Pp, Pq; then

195. PROP. 7. To find the locus of the point of intersection of two tangents to a line of the second order, when the product of the trigonometrical tangents of their inclination to the axis is supposed constant.

Let y=ax+b......(1)

be a straight line which cuts a line of the second order,

referred to the principal diameter, and the tangent at its vertex.

The abscissas of the points of intersection will be found by eliminating y between these equations; we have, therefore,

(ax+b)2= mx + n x2,

or, transposing and arranging,

(a2 — n) x2 + (2 a b — m) x + b2 = 0 . . . . . (3).

Let the points of section be now supposed to coincide; the secant will then become a tangent, and the roots of equation (3) being equal, the equation will be a complete square.

Hence, 4(a2 - n) b2 = (2 a b — m)2,

or m2-4b (am - bn)=0;

therefore, substituting for b its value y-ax, from equation (1), we have

m2 — 4 ( y − a x) { am− n (y—ax)} =0;

therefore, reducing and arranging the result by the powers of a,

.. ny2 — na a′ x2 — maa'x+4m2 = 0 . . . . . . (4).

Therefore, the locus required is an hyperbola or ellipse, according as the constant quantity aa' is positive or negative.

When n=0, or when the curve is a parabola, the equation becomes

therefore, the locus in this case is a straight line, at right angles to the axis.

196. COR. 1. The curve being supposed an ellipse or

hyperbola, let aa'= let aa'==

system of conjugate diameters.

Then equation (4) becomes

or the tangents be parallel to a

Hence, in the ellipse and hyperbola, the locus of the intersection of tangents drawn parallel to a system of conjugate diameters, is an ellipse or hyperbola.

==

197. COR. 2. If aa 1, or the tangents be at right angles to one another,

Y

then, equation (4) becomes

m

y2+x2+ = x + 4 m2 = 0,

n

which is the equation to a circle.

Hence, in the ellipse and hyperbola, the locus of the intersection of pairs of tangents, which form a right angle, is a circle.

whence the locus, in this case, is the directrix.

CHAP. IX.

ON THE GENERAL DISCUSSION OF LINES OF THE

SECOND ORDER.

In this chapter, we shall suppose the axes to be inclined at any angle whatever, unless the contrary be specified.

198. PROP. 1. To find when a line of the second order intersects the axes.

The general equation being

ay2 + bxy+cx2 +dy+ex+f=0,

Let y=0, that is, let the curve cut the axis of x,

Then, cx2+ex+f=0;

1

x=1 ± √(e2 - 4cf),

2

2

which are the abscissas of the points of intersection.

In order, therefore, that the intersection may take place, e' must > 4 cf.

If e2 < 4cf, the values of x are imaginary, and the curve does not meet the axis of x.

If e2=4cf, the two values of x are equal, hence the points of section. coincide, and the curve touches the axis of x.