ANALYTIC GEOMETRY.

CHAPTER I.

THE POINT.

1. THE following method of determining the position of any point on a plane was introduced by Des Cartes in his Géométrie, 1637, and has been generally used by succeeding geometers.

Y

P

N

We are supposed to be given the position of two fixed right lines XX', YY' intersecting in the point O. Now, if through any point P we draw PM, PN parallel to YY' and XX', it is plain that, if we knew the position of the point P, we should know the lengths of the parallels PM, PN; or, vice versa, that if we knew the lengths of PM, PN, we should know the position of the point P.

Suppose, for example, that

we are given PN=a, PM=b,

X'

Y

X

M

we need only measure OM=a and ON=b, and draw the parallels PM, PN, which will intersect in the point required.

It is usual to denote PM parallel to OY by the letter y, and PN parallel to OX by the letter x, and the point P is said to be determined by the two equations x = a, y = b.

2. The parallels PM, PN are called the coordinates of the point P. PM is often called the ordinate of the point P; while PN, which is equal to OM the intercept cut off by the ordinate, is called the abscissa.

The fixed lines XX' and YY' are termed the axes of coordinates, and the point O, in which they intersect, is called the origin. The axes are said to be rectangular or oblique, according as the angle at which they intersect is a right angle or oblique.

It will readily be seen that the coordinates of the point M on the preceding figure are x = a, y=0; that those of the point N are x = 0, y=b; and of the origin itself are x = 0, y = 0.

3. In order that the equations xa, y=b should only be satisfied by one point, it is necessary to pay attention, not only to the magnitudes, but also to the signs of the coordinates.

P

Y

N P

If we paid no attention to the signs of the coordinates, we might measure OM-a and ON=b, on either side of the origin, and any of the four points. P, P1, P P would satisfy the equations x=a, y=b. It is possible, however, to distinguish algebraically between the lines OM, OM' (which are equal in magnitude, but opposite in x' direction) by giving them different signs. We lay down a rule that, if lines measured in one direction be considered as positive, lines measured in the opposite direction must be con

M

P2

N'

Y

M

X

sidered as negative. It is, of course, arbitrary in which direction we measure positive lines, but it is customary to consider OM (measured to the right hand) and ON (measured upwards) as positive, and OM', ON' (measured in the opposite directions) as negative lines.

Introducing these conventions, the four points P, P1, P1⁄2, P ̧ are easily distinguished. Their co-ordinates are, respectively, x=+a) x=a] x=+a) x = − a) y = + b) ' y = + b} ' y = − b) 'y = − b) •

3

These distinctions of sign can present no difficulty to the learner, who is supposed to be already acquainted with trigonometry.

N.B.-The points whose coordinates are xa, y=b, or x=x', y=y', are generally briefly designated as the point (a, b), or the point x'y'.

It appears from what has been said, that the points (+ a, + b), (a, b) lie on a right line passing through the origin; that they are equidistant from the origin, and on opposite sides of it.

4. To express the distance between two points x'y', x"y", the axes of coordinates being supposed rectangular.

and

By Euclid 1. 47,

hence

PQ2=PS" + SQ, but PS= PM- QM' = y' -y",

QS= OM-OM' = x' —x";

I

5. In the following pages we shall but seldom have occasion to make use of oblique coordinates, since formulæ are, in general, much simplified by the use of rectangular axes; as, however, oblique coordinates may sometimes be employed with advantage, we shall give the principal formulæ in their most general form.

Suppose, in the last figure, the angle YOX oblique and

=w, then

and

PSQ=180°-w,

PQ2 = PS" + QS-2PS. QS.cos PSQ,

or, PQ2 = (y' — y'')2 + (x′ — x")2 + 2 (y' — y') (x' — x') cos w.

Similarly, the square of the distance of a point, x'y', from the origin="+y" + 2x'y' cos w.

In applying these formula, attention must be paid to the signs of the coordinates. If the point Q, for example, were in the angle XOY', the sign of y" would be changed, and the line PS would be the sum and not the difference of y' and y". The learner will find no difficulty, if, having written the coordinates with their proper signs, he is careful to take for PS and QS the algebraic difference of the corresponding pair of coordinates.

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Ex. 1. Find the lengths of the sides of a triangle, the coordinates of whose vertices are x = 2, y′ = 3; x′′ = 4, y'' = − 5 ; x′′" — — 3, y'" — — 6, the axes being rectangular. Ans. 168, 150, 106.

Ex. 2. Find the lengths of the sides of a triangle, the coordinates of whose vertices are the same as in the last example, the axes being inclined at an angle of 60°. Ans. J52, 57, √151.

Ex. 3. Express that the distance of the point xy from the point (2, 3) is equal Ans. (x2)2 + (y-3)2 = 16

to 4.

Ex. 4. Express that the point xzy is equidistant from the points (2, 3), (4, 5).
Ans. (x2)2 + (y − 3)2 = (x − 4)2 + (y-5)2; or x + y = 7.

Ex. 5. Find the point equidistant from the points (2, 3), (4, 5), (6, 1). Here we have two equations to determine the two unknown quantities x, y.

6. The distance between two points, being expressed in the form of a square root, is necessarily susceptible of a double sign. If the distance PQ, measured from P to Q, be considered positive, then the distance QP, measured from Q to P, is considered negative. If indeed we are only concerned with the single distance between two points, it would be unmeaning to affix any sign to it, since by prefixing a sign we in fact direct that this distance shall be added to, or subtracted from, some other distance. But suppose we are given three points P, Q, R in a right line, and know the distances PQ, QR, we may infer PR = PQ+ QR. And with the explanation now given, this equation remains true, even though the point R lie between P and Q. For, in that case, PQ and QR are measured in opposite directions, and PR, which is their arithmetical difference, is still their algebraical sum. Except in the case of lines parallel to one of the axes, no convention has been established as to which shall be considered the positive direction.