Page images
PDF
EPUB
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][merged small][merged small]

Students reading this work for the first time may omit Chapters

IV, VII, XIV, XV, XVI.

PLANE CO-ORDINATE GEOMETRY.

CHAPTER I.

CO-ORDINATES OF A POINT.

R 1. IN Plane Co-ordinate Geometry we investigate the properties of straight lines and curves lying in one plane by means of co-ordinates; we commence by explaining what we mean by the co-ordinates of a point.

[blocks in formation]

Let O be a fixed point in a plane through which the lines ΧΟΧ, X'OX, Y'OY, are drawn at right angles. Let P be any other point in the plane; draw PM parallel to OY meeting OX in M, and PN parallel to OX meeting OY in N. The position of P is evidently known if OM and ON are known; for if through N and M lines be drawn parallel to OX and OY respectively, they will intersect in P.

The point O is called the origin; the lines OX and OY are called axes; OM is called the abscissa of the point P; and.

T. C. S.

1

ON, or its equal MP, is called the ordinate of P. Also OM and MP are together called co-ordinates of P.

2. Let OM-a, and ON=b, then according to our definitions we may say that the point P has its abscissa equal to a, and its ordinate equal to b; or, more briefly, the co-ordinates of the point P are a and b. We shall often speak of the point which has a for its abscissa and b for its ordinate, as the point (a, b).

3. A distance measured along the axis OX is however most frequently denoted by the symbol x, and a distance measured along the axis OY by the symbol y. Hence OX is called the axis of x, and OY the axis of y. Thus x and y are symbols to which we may ascribe different numerical values corresponding to the different points we consider, and we may express the statement that the co-ordinates of P are a and b, thus; for the point P, x = a and y = b.

[ocr errors]

4. The lines X'OX, Y'OY, being indefinitely produced divide the plane in which they lie into four compartments. It becomes therefore necessary to distinguish points in one compartment from points in the others. For this purpose the following convention is adopted, which the reader has already seen in works on Trigonometry; lines measured along OX are considered positive and along OX' negative; lines measured along OY are considered positive, and along OY' negative. (See Trigonometry, Chap. IV.) If then we produce PN to a point Q such that NQNP, we have for the point Q, xa, y=b. If we produce PM to R so that MR MP, we have for the point R, x=a, y=-b. Finally if we produce PO to S so that OS OP, we have for the point S, x= == -a, y: -b.

=

=

5. In the figure in Art. 1 we have taken the angle YOX a right angle; the axes are then called rectangular. If the angle YOX be not a right angle, the axes are called oblique. All that has been hitherto said applies whether the axes are rectangular or oblique. We shall always suppose the axes rectangular unless the contrary be stated; this remark applies both to our investigations and to the examples which are given for the exercise of the student.

POLAR CO-ORDINATES OF A POINT.

3

6. Another method of determining the position of a point in a plane is by means of polar co-ordinates.

[merged small][ocr errors]
[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][subsumed][subsumed][subsumed][subsumed][subsumed]

P

Let O be a fixed point, and OX a fixed line. Let P be any other point; join OP; then the position of P is determined if we know the angle XOP and the distance OP. The angle is usually denoted by 0 and the distance by r.

O is called the pole, OX the initial line; OP the radius vector of the point P, and POX the vectorial angle.

7. The position of any point might be expressed by positive values of the polar co-ordinates @ and r, since there is here no ambiguity corresponding to that arising from the four compartments of the figure in Art. 4. It is however found convenient to use a similar convention to that in Art. 4; angles measured in one direction from OX are considered positive and in the other negative. Thus if in the figure XOP be a positive angle, XOQ will be a negative angle; if the angle XOQ be a quarter of a right angle, we may say that for XOQ, 0=-. It is, as we have stated, not absolutely necessary to introduce negative angles, but convenient; the position of the line OQ, for instance, might be determined by measuring from OX in the positive direction an angle =2π- as well as by measuring an angle in the negative direction=.

π

8

π

8

Also positive and negative values of the radius vector are admitted. Thus, suppose the co-ordinates of P to be

π

4

and a,

[ocr errors]
« PreviousContinue »