ANALYTICAL GEOMETRY

OF

TWO DIMENSIONS.

If we reflect on the nature of Geometrical Problems, we shall perceive that the greater number of them depend ultimately on finding the distance of one or more unknown points, from other points or straight lines, whose position is already known and determined. If, therefore, we have a method which enables us to determine analytically the position of a point, with reference to certain other points or straight lines whose position is known, we shall be in a state to resolve all kinds of geometrical problems.

Let there be two straight lines AX, AY, whose position is known and determined, situated in the same plane at right angles to each other, and let P be any point in the same plane whose position we are required to determine.

N

Y

P

A

M

X

From the point P let fall PM, PN, perpendiculars on AX and AY. Then it is manifest that the point P will be determined, if we know the length of the sides AM, AN, of the rectangle AP. For these sides are the distances of the point P from the two fixed straight lines AX, AY, so that, if we draw from the points M and N two straight lines, respectively parallel to AY and AX, the point where they intersect will be the point required.

The two fixed lines AX, AY are called Axes.

The distance AM or PN of the point P from the axis AY is called the Abscissa of the point P, and is usually designated algebraically by the letter x. The distance AN or PM of the point P from the axis AX is called the Ordinate of the point P, and is usually designated algebraically by the letter y.

The two distances x and y are together denominated the Co-ordinates of the point P.

The two axes are distinguished from each other by calling the axis AX, along which the abscissas are reckoned, the Axis of Abscissas, or the Axis of r's; and in like manner the axis AY, along which the ordinates are reckoned, is called the Axis of Ordinates, or the Axis of y's.

The point A is called the Origin of Co-ordinates, since it is from this point that the distances are reckoned.

EQUATIONS OF A POINT.

The characteristics of every point situated on the axis of y's is x = 0, since that equation indicates that the distance of the point in question from that axis is nothing.

Similarly the characteristic of every point situated on the axis of x's is y=0

Hence the system of two equations, x = 0, y = 0, characterizes the point A the origin of co-ordinates, since these equations can hold good at the same time for no other point.

In general the two equations x = a, y = b, when considered together characterize a point situated at a distance a from the axis of y's, and at a distance b from the axis of x's. The first of these equations, when considered separately, belongs to all the points of a straight line drawn parallel to the axis of y's, at a distance AM = a, and the second to all the points of a straight line drawn parallel to the axis of x's, at a distance AN = b. Hence the system of two equations together belongs to the point P, in which these lines intersect, and belongs to this point alone. These expressions are thus, as it were, the analytical representations of the point, and for this reason are called the Equations of the point.

We must always consider, in the expressions a and b, not only the absolute or numerical values of the distances of the point from the two axes, but likewise the signs by which they may be affected, according to the position of the point in the plane of the axes AX and AY. For, according to the conventions explained in the first chapter of Analytical Plane Trigonometry, if we agree to consider as positive, distances such as AM reckoned along AX to the right of the point A, we ought to consider as negative, distances such as AM' reckoned to the left of the same point.

P'

X' M'

Y

P

N

X

All the other remarks which we have made upon the supposition that the axes were rectangular, apply equally to the case in which they are oblique.

X'

M'

A

Y'

Y

N

P

In order to complete our discussion on the equations to a point, let it be required

To determine the analytical expression for the distance between two given points which are situated in the same plane.

Let the co-ordinates of the first point P, be x', y', and of the second point P, be x', y', so that the equations to these points, whose positions we suppose known, are

Of P, {

y = y'

}.....(1) And of P, {

Sx = x"
y = y"

} ....(2)

It is required to express the distance P, P, of these points in terms of the given co-ordinates x', y', x", y".

But P,Q M2M, = AM, AM,
And P,Q = P,M, QM2 = P,M,

Substituting these values of P,Q and P,Q, in (A) we have

Rx") + (y — ye

Ꭱ =

This formula is quite general, and will apply equally well to the case in which the two points are situated on different sides of the axes.

It will only be necessary, in this case, to introduce the changes in the signs which correspond to changes in position; thus, for example, to obtain the distance of two points, one of which is situated in the angle YAX, and the other P, in the angle YAX', we must change the sign of a", which gives us

R=√(x+1)2 + (y' — y'')2

P:

X

M2 AM

X

In fact, if we perform the calculations as in the former case, we find

And

2

R = √(x+x")° + (y′ — y'')'

If one of the points, P, for example, is the origin of co-ordinates, in that case "= 0, y' 0, and the formula becomes

P,, . . . . . and from these points draw P,M,, P,M,, P ̧M,, . . . . . perpendicular on AX, and through the point O, in which the straight line meets the axis AY, draw OQ parallel to AX.

The similar triangles P,Q1O, P,Q,O, P,Q¿O, will give a series of equal ratios.

=

= , or, since AO = Q1M1 = Q,M2 = &c. Q2O Q2O Q30

P1M1 AO P2M, АО P.M. AO

AM,

=

AM,

= &c.

Which proves that the difference between the ordinate of any point in a straight line, and the distance of the straight line from the origin, is in a constant ratio to the abscissa of the same point.

Let us then call the co-ordinates of any point in the straight line x and y, and let us designate by b the distance AO; that is, the distance from the origin of the point in which the straight line cuts AY; let a be the constant ratio which we have just mentioned, we shall then have the relation.

Or,

Now this relation holds good, as has been shown above, for every point in the straight line LOS, but it will not hold good for any point which is not situated in this straight line.

For let N be any other point taken either above or below the straight line LOS.

Now, since the ordinate NPM of that point is either greater or less than the ordinate PM of the straight line corresponding to the same abscissa AM, and since by hypothesis we have for the point P the relation

N'

M X

We thus perceive that the relation (1) is characteristic of every point in the straight line LOS, and that it does not hold good for any point without that line, and is therefore the analytical representation of that straight line; for if this relation be given in the first instance, we are enabled, by means of it, to determine the position of the straight line, and to trace it graphically.

For this purpose it is sufficient to give to a a series of values, which we measure along AX, such as AM1, AM,, &c. and drawing from these points straight lines M,P,, M,P2, parallel to AY, and making these straight lines equal to the corresponding values of y, found from equation (1), we shall in this manner determine the points P1, P2, ..... situated in the required straight line.

PZ

Q1 Q2 Q3 Q

A M1 M2 M3 X

For this reason the relation (1) is denominated the Equation to the Straight Line LOS.