PROB. IX.

-To find the equation to the circle, the axes of co-ordinates being inclined at any angle.

Let the straight lines AX, AY, which are inclined to each other at a given angle Q, be assumed as axes.

Take P any point in the circle, and let the co-ordinates of P be called x and y.

Let C be the centre of the circle, and let the co-ordinates of the point C be x' y'. Draw PM, CM', parallel to AY; and PQ, CQ parallel to AX; produce Q'C to meet PM in N; join C, P;

Q'

M' M

X

CP=r, AM = x, MP = y, AM' = x', CM' = y', <e YAX = &

2NP. CN cos. CNP.

MP = AM

- YAX) = (180° — q).

Hence the above equation becomes

r2 = (y- y)2 + ( x— x')° + 2 (x — x) (y — y') cos.

which is the equation required.

Before proceeding farther with this subject, it may not be improper to make some general observations on the nature of equations to lines, and on the use which we may make of them.

We have seen that the position of a straight line and of a circle is fixed upon a plane by means of an equation between the co-ordinates of each of its points, and a certain number of constant quantities; the knowledge of which enables us to determine its position geometrically.

Suppose then, that x and y being considered to denote the distance of a point from two rectangular or oblique axes, the resolution of some problem has led to an equation of the form

f(x, y) = 0.

If we wish to fix the position of the point which verifies this equation, we shall find that there are an infinite number of such points, and that this series of points constitutes a line which is either straight or curved according to the nature of the equation. For since there is only one equation between x and y we may give any value we please to one of these variables, and then the equation will give the corresponding value of the other variable.

Let

us, for example, give the abscissa x the series of values

....

If y enters in the equation in the simple power only, we shall derive from the equation a succession of corresponding values of that variable;

y = b1, bg, bз, b1.

If in AX we take AM1, AM,, AM, AMĄ,

equal to a, a,, αз, ɑs, ・・ and from the points

M1, M., M., M., .... raise perpendiculars M,P1,

M.P., M.P., M.P1,

ხი, ხვ, ხ,

equal respectively to b1,

we shall find a series of points P1,

P2, P3, P.; all of which will equally satisfy the conditions of the equation.

Now, since we may give to x a series of values

which differ very little from each other, and since, in that case, the successive values of y will, generally speaking, likewise differ very little from each other; the points P1, P、, PË, P1, will be very near to each other, and we shall

be thus enabled to unite all these points with each other by means of a con

...

tinuous line, P1, P′, P3, P4, . . . . all the points of which will be so many solutions of the question, since all the intermediate points of this line comprised between those constructed in the manner described, may be supposed to correspond to the values of x and y derived from the equation of the problem.

Ꮖ

The form of the curve will be determined with more accurate precision in proportion, as the points P1, P2, P3, P4, . . . . are nearer to each other. Let us now suppose that y enters into the proposed equation in some power higher than the first. Since in this case, for each value of x, there will be two or more corresponding values of y, according to the degree of the equation; it follows that the curve will be composed of two or more branches, P1, P2, P3, Q1, Q2, Q8,

....

R1, R2, R3,

....

Let it be required, for example, to construct the curve whose equation is

y2 = 2x

Solving the equation for y, we have

y = + √2x.

Ri

R2

R3

Y

QI

Q2

૨૩

P3

Which proves, in the first place, that for each value of x there are two corresponding values of y equal to each other, but with opposite signs; and in the second place, that for all negative values of x the corresponding values of y are imaginary, that is to say, that the curve can have no point situated to the left of the origin AY,

This being established, let us make x = 0, then y =0, which shows that the origin of co-ordinates is placed on some point in the curve; or, in other words, that the curve passes through the origin.

Next, let x = 1 .. y = ± √2 =+ 1.4

Take therefore upon AX a distance AM1 = the linear unit; and from M, draw a perpendicular to AX, and on each side of AX take M, P1, M1 p1 = 1.4. ...; then P and p will be two points in the required curve.

Next, let x 2 y = + √T=+2 Constructing these values as before, we shall find P„, and p„, for two new points in the curve.

Continuing in this manner to give a succession of values to x, and construct. ing the corresponding value of y, we shall obtain a curve of the form VAv, which consists of two branches AV, and Av, which extend indefinitely to the right of AY, since for all positive values of x, the corresponding values of y are real.

For a second example, let us take the equation

We perceive, in the first place, that, for the same value x, there are two equal values of y with contrary signs; and, in the second place, that, whatever value

T T

we give to x, whether positive or negative, we shall always obtain real values for y. Hence we can conclude at once, that the curve extends indefinitely both above and below the axis AP, and both to the right and left of AY. Let us now make some particular suppositions.

= +2.

Take on AY two distances AB, Ab, each 2, the points B and b belong to the curve.

On AX take AM1 = 1 through M, draw a straight line P1 p1 parallel to AY, and make M, P1 = M1 p1 = 2.2

in the curve,

Let x =

then P, and p, are two new points

y= ±√8 = 2.8

....

Constructing this value of y in the same manner, we obtain P, and p, for two other points in the curve, and so on for the other points to the right of AY. In order to obtain the points to the left of AY, since the values of x, which are numerically the same but taken with different signs, correspond to the same values of y; it will be sufficient to take Am, Am, equal to AM1, AM2, . . . . and through the points m1, m1, to draw straight lines parallel to AY, and through the points P1, P1, P2, P1, straight lines parallel to AX, and we shall thus determine the points Q1, 91, Q2, qa, belonging to the curve, which will evidently be composed of two branches distinct and opposite, P2BQ2 P2 b q2

....

The curve represented by an equation between x and y, is called the Geometrical Locus of the equation.

Reciprocally, if a curve be traced upon a plane, and if by any means founded upon the definition or upon some characteristic property of the curve, we can arrive at a relation which exists between the co-ordinates x and y of all points in that curve, and exists for these points alone; the relation thus obtained is called the Equation to the Curve.

We shall now proceed in this manner to obtain the equations to the most important curves.

PROB. X.

To find the equation to the parabola.

DEFINITION.—A Parabola is the locus of a point whose distance from a given fixed point, and from a straight line given in position, is always the same. Let S be the given fixed point, and Nn the straight

line given in position;

Draw SK perpendicular to Nn, and bisect SK in A ;
Then by definition A is a point in the parabola.

Take P any point in the curve and join S, P;

From P draw Pm perpendicular to Nn;

From A draw AY perpendicular to ASX.

Let A be the origin and AX, AY, the axes of coordinates.

From P draw PM perpendicular on AX.

Equating these two values of r2 we obtain a relation between x and y. y2 + (x — m)2 = (x + m)2

which is the equation to the parabola.

In order to find the value of the ordinate passing through the focus

which shows that 4m is the double ordinate passing through the focus, or the Latus Rectum of the parabola.

Solving the equation for y

y = ± 2√mx

For all negative values of x, y is impossible, which shows that there is no point of the curve to the left of the origin A.

Which shows that the curve passes through the origin, as is evident from other considerations.

Giving a succession of positive values to x, we perceive that as x increases, y increases also, and that for each value of x there will be two equal values of y with opposite signs.

Hence the curve extends indefinitely to the right of A, and is symmetrically situated with regard to AX.

PROB. XI.

To find the equation to the Ellipse.

DEFINITION.-An Ellipse is the locus of a point, whose distance from two given fixed points is equal to a constant quantity.

Let S and H be the two given fixed points;

Join S, H; bisect SH in C;

Let P be any point in the curve, join S, P; H, P ;

Draw PM perpendicular to CX;

Draw CY perpendicular to HS, let C be the origin, and CX, CY, the axes of co-ordi

nates.

Let the quantity, to which the sum of SP and HP is always equal, be 2a.

X a H

Y

Then, let CM = x, MP = y, SP =r, HP = r', SH = 2c.
Then we have

If, therefore, we eliminater and r' between these three equations, we shall obtain a relation between x and y, which will be the required equation to the

Sincer and r' or 2a is always SH or 2c, .. a is always c, and .. the quantity a2c2 is essentially positive.

2c2x2
a2

which is the most simple form of the equation to the ellipse.

Solving the equation for y and x in succession, we obtain

y3

Xa