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CHAPTER IX.

ON THE TANGENTS, NORMALS AND ASYMPTOTES TO CURVES.

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If the equation to the curve be put under the form

y = f(u),
the equation to a tangent at a point ry

is
dy
y' - y = (2' – x);

dc
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

du
(u' – x) + (x - 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

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or

(- y)

du

(x' - x) = 0.

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The perpendicular from the origin on the tangent is

du

ty ydir

wdy d x dy p = (dx? + dy')

du +

+

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

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

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

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dy

dr y

along the axis of x. These I shall call yo, x, respectively.

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

X Y = m.
du du
Then

Y, X, and the equation to the tangent is da

y (a - c) + (g - y) = 0;

or yx' + xy' = 2xy = 2 m.

dy

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as X Y = m.

am? The perpendicular on the tangent p =

(t° + y)

m?

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Also Y. = y +
24, and = x + = 27.

Y
Hence the product of the intercepts of the tangent
EY. = 4X Y

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

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

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Hence the equation to the tangent is

' Ý'

(ax)! (by)
The intercepts are 2, = (ax)\, y = (by);

= 1.

therefore + - + = 1;

or x,, Yo 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

2+ y = a. The equation to the tangent is

ze' ý

T+ 23

as

Therefore x, = x, y, = 3

aša), y, = aa y' ; and the portion of the tangent intercepted between the axes = (.r." + y;-) = a; or the hypocycloid is constantly touched by a straight line of given

The con

length which slides between two rectangular axes. verse 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. II. p. 447.)

For the perpendicular from the origin on the tangent we find

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

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(5) In the logarithmic curve

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

y
The subnormal = The normal = (a +9°)%.

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a

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cy
(y - c)}"

The tangent =

(y - z)^^

() From the general parabolic equation

y" = a"-"X, we find the equation to the tangent to be

m x (' y) = y (x' – x).

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.

T-

No
Subtangent

· y
Y
Yo

Y
(9) The equation to the cycloid referred to its vertex is

dy 2 a

dx AB (fig. 19) being the axis of a.

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

MN (20 x – 2?) dy
tan NANE
AN

dx 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 JB. 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 x – xr")! = PN - MIN = PM. But from the generation of the curve, PM is equal to the arc of the circle AV, therefore yo

= arc AM.

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