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The equation to a tangent at the point (X124) is

XY. + xy=} (a? + b). The area of a triangle contained between these axes and a tangent is constant, and =} (a + b) sin any.

Let (x1,9.) be any point in the curve, and a, b, the conjugate diameters to that point, then

a,= + 2(x, y.)* cos x,y; by=+2(x1 y.)sin x,y.
If UWu is parallel to Ci, then UW=Wu.
(H.C. S. 245—59; Biot, 250–8; Hust. Prop. 11–5.)

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20

DISCUSSION OF LINES OF THE SECOND ORDER. (21.) The general form of an equation of the second degree containing two unknown quantities

ay® + bxy + c#? + dy +ex+f=0. From the solution of this equation

bx+d 1 y=

+

(bo – 4ac).* + 2(bd 2 ae) x + ď* — 4af\"; 2 a by te 1

+ - (bo 4ac)yo +2(be 2cd)y + e* – 4cf|*.

2c which values of w and y may be thus represented :

6x + d 1 y=

+ Ax* +2Bx+C); 2a

2a by te 1

Ay' + 2 By+C

2B,y+C)*: 2c 2c A straight line cannot cut a line of the second order in more than two points.

The locus of the middle points of any number of parallel chords is a diameter.

If the chords drawn parallel to any diameter are bisected by another diameter, then the chords parallel to the latter, will be bisected by the former. The co-ordinates of the centre of the curve are 2 ae-bd

2cd-be 62 - 4ac

1° -4ac

+

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If bo 4ac<0, the curve is an ellipse ;
h - 4ac>0,

: an hyperbola ;
- 4ac=0,.

a parabola. If b=0, a and c must have the same sign in the ellipse, and different signs in the hyperbola.

If the curve is an ellipse or hyperbola, and the axes parallel to any system of conjugate diameters, the form of the equation is

ay' + cx? + dy +ex+f=0. If the origin is at the centre of the curve, the equation becomes

ay' + bxy +cm? +f=0: and if the curve is also referred to conjugate diameters, then

ay' +00? +f=0. If the axes are parallel to the asymptotes of an hyperbola, the form of the equation is

bxy + dy + ex+f=0. If e'>4cf, the curve intersects the axis of æ in two points ;

e = 4cf, . .. touches the axis of x;

e* <4cf, ....... does not meet the axis of x; The same conditions exist with regard to the axis of y, according as d? >, =, or < 4af.

bxtd If Bo>AC, the curve intersects the diameter, y=

2a

Bo=AC,
- .

touches that diameter;
B< AC,

does not meet it.
(22.) Determination of the species, and their varieties.

[1] If A<0, the curve is an ellipse.
If the roots of the equation

2B C

X + 0, (a)

А A are equal, the ellipse is reduced to a point.

If the roots are impossible, the ellipse becomes an imaginary line.

If a=c, and b=0, the ellipse becomes a circle.

x2 +

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.

[2] If A>0, the curve is an hyperbola.
If the roots of the equation (a) are possible, the diameter,

bx + d
y=

2c cuts the curve; if impossible, that diameter does not meet it.

If the roots are equal, the hyperbola becomes two intersecting straight lines.

If a= -c, and b=0, the hyperbola is equilateral.
[3] If A=0, the curve is a parabola.

If B also =0, and
C>0, the equation represents two parallel straight lines,
C=0,

. one straight line, C<0,

an imaginary line. (H. A. G. 81–89, 184—213; G. G. A. Ch.iv; Biot, 264–304.)

SUMMARY OF EQUATIONS. (23.) Rectilinear equations.

[1] Let the curve be referred to its centre, and principal diameter, then

62 in the ellipse,

yo = (ao — w?);

62
hyperbola,
. circle,

yo=a— X; equilateral hyperbola, y = x -a. [2] Let the curve be referred to the principal diameter, and a tangent at the vertex, then

62 in the ellipse,

(2ax - x); а

62 . hyperbola,

yo : (2ax + xo);

a? parabola,

= 4mx ; circle,

y=2a x - x; equilateral hyperbola, yo = 2ax + x2. The general form of the equation is =mx+ nao.

=

a

у°

.

az (x2 – a);

.

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u=

The same equations subsist, if a and b are any system of conjugate diameters to which the curve is referred. (24.) Polar equations. [1] Let the curve be referred to the radius vector, and angle contained between it, and the principal diameter, then 1. The centre being the pole ;

aʼ(1 - e) in the ellipse,

1-(e.cos )?

(e“ – 1) . hyperbola, u’=

(e.cos ) – 1 2. The focus being the pole ;

a (1 - e) in the ellipse,

1+e.cos ?

a(e“ – 1) . hyperbola, u=

1+ e cos

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2 m

parabola, u=

1 + cos e [2] Equations between the radius vector, and a perpendicular on the tangent from the pole. 1. The centre being the pole :

a 62
in the ellipse,

pe =
ao +5° do?

a’72
. hyperbola,

alo Cao b°)

a . equilateral hyperbola, p

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ANALYTICAL GEOMETRY OF THREE DIMENSIONS.

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(25.) The straight line. The equations to a straight line referred to three rectangular co-ordinates, are

X = ax + a, y=bx + ß: which are the equations to its projections on the planes of xz, yx respectively.

The equations to a line passing through the point (X1,41,7ı), are

X X,=a(z— x,), y-Y.=b (x—%,). The equations to a line passing through two points (R1,91%), (x,y,z), are

Y2 - Y1 X X 15

y-Yi=

(5—X,). %, - %

%, — %1
In order that the two lines
fo=0% +012

X = 0,% + Ang
ly=b,x+Bz;

v=box+3, may intersect each other, it is necessary that

ai ag

B.-B,

01 - A2 bi b,
The co-ordinates of the point of intersection are

QA2 - 0201 biß, - baß, az
01 A2

01 -a,

01-02 The distance (d) of a point (X13910%,) from the origin ;

d,=(x," + y + x,%)}= (1 + a2 + b)}; if w=ax, y=bx are the equations to the line passing through the given point, and the origin. The distance (D) between two points (@v3413%), (x2392,7.) ;

D=(x, — x,) + (y1 - y.)” + (,-2)):

=d," + d. – 2(27X2 + yıy2 + %1%)| d, being the distance of (x2392,%,) from the origin.

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