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CHAPTER IX.

ON THE TANGENTS, NORMALS AND ASYMPTOTES TO CURVES.

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du

If the equation to the curve be put under the form

y = f (x),
the equation to a tangent at a point xy is

dy
y' - y (a' - 2);

der
a' and y' being the current co-ordinates of the tangent.
If the equation to the curve be put under the form

u = Q(x, y) = C, the equation to the tangent takes the more symmetrical form

du
(V' - x) + (y' - y) = 0.
dx

dy If u be a homogeneous function of n dimensions in x and y, by a well-known property of such functions

du +y

dy
and the equation to the tangent becomes

du du
+ y

dy
The equations to the normal are

dc

(x' - x);

dy
du

du
(y - y) (x' - x) = 0.
d.c

dy

du die

= nu = nc,

x

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d x

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or

The length of the subtangent is y

dx dy dy

The length of the subnormal is y

dx

idx) The length of the tangent is +

dy

dy The length of the normal is +

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dx

nc

idu

2

2

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The perpendicular from the origin on the tangent is

du du

+ Y ydx - ady

dy p = (dx* + dys)!

du

du +

+ dy

dy if u be a homogeneous function of n dimensions in w and y.

The portion of the tangent intercepted between the point of contact and the perpendicular on it from the origin is

du du

y
wdx + ydy dy d x
t
(dx? + dy')} du ? du

dy The portions of the axes cut off between the origin and the tangent, or the intercepts of the tangent, are

dy Y - X

along the axis of y. dx

+

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These I shall call y, x, respectively.

Ex. (1). The equation to the hyperbola referred to its asymptotes is

X Y = m*.

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du

= x, and the equation to the tangent is

dy

y (a' – x) + x (y' - y) = 0;
or yx' + xy' = 2 xy = 2 m2.

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=

dy
m?

da
Since
the subtangent

=-X,

- y dx

dy ma as X Y = ma.

2 m2 The perpendicular on the tangent p =

(x® + yo)!

m?

m2

Also
Yo = y + 2y, and a = x +

= 2x.

y
Hence the product of the intercepts of the tangent

X, Y.
4xY

4 ma is constant ; and the triangle contained between the axes and the tangent, being proportional to this product, is also constant.

(2) The equation to the parabola referred to two tangents as axes is

y

2
= 1.

+

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+

= 1.

(by)

Hence the equation to the tangent is

x y'

(ax)?
The intercepts are Xo = (ax)\, % = (by)!;

2
therefore

= 1;

b or X, Y, are the co-ordinates of the chord joining the points at which the axes touch the curve.

(3) The equation to one of the hypocycloids referred to rectangular co-ordinates is

x} + y} = a). The equation to the tangent is

x ý

as.

y Therefore ~, = aš rl, y, = ašys; and the portion of the tangent intercepted between the axes = (x + + y.3)} = a; or the hypocycloid is constantly touched by a straight line of given

+

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length which slides between two rectangular axes. The converse of this proposition, viz. that the locus of the ultimate intersections of a line of given length sliding between rectangular axes is this hypocycloid, was first shewn by John Bernoulli. (See his Works, Vol. 1. p. 447.)

For the perpendicular from the origin on the tangent we find

p = (a xy). (4) In the cissoid of Diocles,

X3 y?

;

2a - X

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The subtangent = a, and is therefore constant.
The tangent = (a + yo).

yo
The subnormal The normal = (ao + yo)4

.

.

a

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The tangent

(y - c*)?

(7) From the general parabolic equation

y = am-1x, we find the equation to the tangent to be

mx (y' y) = y (x' x).

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.

x2

NO
Subtangent

y

Xo
y
yo

y
The equation to the cycloid referred to its vertex is
dy

a)

(9)

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AB (fig. 19) being the axis of x.

If M be the point where the ordinate meets the generating circle, and if we join MA, MB, then

MN (20 x – ?)? dy
tan MAN-
AN

dix That is to say, the tangent to the cycloid is parallel to the chord of the generating circle. The normal is evidently parallel to the other chord MB. Hence also the angle which two tangents make with each other is equal to the angle between the corresponding chords of the generating circle.

y = Y - (2a w – xo)! = PN - MN = PM. But from the generation of the curve, PM is equal to the arc of the circle AM, therefore y. = arc AM.

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