taken at random in the plane will, as a rule, not satisfy the equation, but infinitely many points, and in most cases infinitely many real ones, have coordinates which do satisfy it, and these points are exactly those which lie upon some locus of one dimension, a straight line or more frequently a curve, which is said to be represented by the equation. Take, for instance, the equation y = mx, where m is a given constant. It is satisfied by the coordinates of every point P, which is such that, in fig. 48, the distance MP, with its proper sign, is m times the distance OM, with its proper sign, i.e. by the coordinates of every point in the straight line through O which we arrive at by making a line, originally coincident with x′Ox, revolve about O in the direction opposite to that of the hands of a watch through an angle of which m is the tangent, and by those of no other points. That line is the locus which it represents. Take, more generally, the equation y = φ(x), where φ(x) is any given non-ambiguous function of x. Choosing any point M on x′Ox in fig. 1, and giving to x the value of the numerical measure of OM, the equation determines a single corresponding y, and so determines a single point P on the line through M parallel to y ′Oy. This is one point whose coordinates satisfy the equation. Now let M move from the extreme left to the extreme right of the line x′Ox, regarded as extended both ways as far as we like, i.e. let x take all real values from −∞ to ∞. With every value goes a point P, as above, on the parallel to y ′Oy through the corresponding M; and we thus find that there is a path from the extreme left to the extreme right of the figure, all points P along which are distinguished from other points by the exceptional property of satisfying the equation by their coordinates. This path is a locus; and the equation y = φ(x) represents it. More generally still, take an equation ƒ(x, y) = 0 which involves both x and y under a functional form. Any particular value given to x in it produces from it an equation for the determination of a value or values of y, which go with that value of x in specifying a point or points (x, y), of which the coordinates satisfy the equation ƒ(x, y) = 0. Here again, as x takes all values, the point or points describe a path or paths, which constitute a locus represented by the equation. Except when y enters to the first degree only in ƒ(x, y), it is not to be expected that all the values of y, determined as going with a chosen value of x, will be necessarily real; indeed it is not uncommon for all to be imaginary for some ranges of values of x. The locus may largely consist of continua of imaginary points; but the real parts of it constitute a real curve or real curves. Note that we have to allow x to admit of all imaginary, as well as of all real, values, in order to obtain all imaginary parts of the locus.
A locus or curve may be algebraically specified in another way; viz. we may be given two equations x = ƒ(θ), y = F(θ), which express the coordinates of any point of it as two functions of the same variable parameter θ to which all values are open. As θ takes all values in turn, the point (x, y) traverses the curve.
It is a good exercise to trace a number of curves, taken as defined by the equations which represent them. This, in simple cases, can be done approximately by plotting the values of y given by the equation of a curve as going with a considerable number of values of x, and connecting the various points (x, y) thus obtained. But methods exist for diminishing the labour of this tentative process.
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| Fig. 50. |
Another problem, which will be more attended to here, is that of determining the equations of curves of known interest, taken as defined by geometrical properties. It is not a matter for surprise that the curves which have been most and longest studied geometrically are among those represented by equations of the simplest character.
8. The Straight Line.—This is the simplest type of locus. Also the simplest type of equation in x and y is Ax + By + C = 0, one of the first degree. Here the coefficients A, B, C are constants. They are, like the current coordinates, x, y, numerical. But, in giving interpretation to such an equation, we must of course refer to numbers Ax, By, C of unit magnitudes of the same kind, of units of counting for instance, or unit lengths or unit squares. It will now be seen that every straight line has an equation of the first degree, and that every equation of the first degree represents a straight line.
It has been seen (§ 7) that lines parallel to the axes have equations of the first degree, free from one of the variables. Take now a straight line ABC inclined to both axes. Let it make a given angle α with the positive direction of the axis of x, i.e. in fig. 50 let this be the angle through which Ax must be revolved counter-clockwise about A in order to be made coincident with the line. Let C, of coordinates (h, k), be a fixed point on the line, and P(x, y) any other point upon it. Draw the ordinates CD, PM of C and P, and let the parallel to the axis of x through C meet PM, produced if necessary, in R. The right-angled triangle CRP tells us that, with the signs appropriate to their directions attached to CR and RP,
and this gives that
an equation of the first degree satisfied by x and y. No point not on the line satisfies the same equation; for the line from C to any point off the line would make with CR some angle β different from α, and the point in question would satisfy an equation y − k = tan β(x − h), which is inconsistent with the above equation.
The equation of the line may also be written y = mx + b, where m = tan α, and b = k − h tan α. Here b is the value obtained for y from the equation when 0 is put for x, i.e. it is the numerical measure, with proper sign, of OB, the intercept made by the line on the axis of y, measured from the origin. For different straight lines, m and b may have any constant values we like.
Now the general equation of the first degree Ax + By + C = 0 may be written y = −(A/B)x − C/B, unless B = 0, in which case it represents a line parallel to the axis of y; and −A/B, −C/B are values which can be given to m and b, so that every equation of the first degree represents a straight line. It is important to notice that the general equation, which in appearance contains three constants A, B, C, in effect depends on two only, the ratios of two of them to the third. In virtue of this last remark, we see that two distinct conditions suffice to determine a straight line. For instance, it is easy from the above to see that
| x | + | y | = 1 |
| a | b |
is the equation of a straight line determined by the two conditions that it makes intercepts OA, OB on the two axes, of which a and b are the numerical measures with proper signs: note that in fig. 50 a is negative. Again,
| y − y1 = | y2 − y1 | (x − x1), |
| x2 − x1 |
i.e.
represents the line determined by the data that it passes through two given points (x1, y1) and (x2, y2). To prove this find m in the equation y − y1 = m(x − x1) of a line through (x1, y1), from the condition that (x2, y2) lies on the line.
In this paragraph the coordinates have been assumed rectangular. Had they been oblique, the doctrine of similar triangles would have given the same results, except that in the forms of equation y − k = m(x − h), y = mx + b, we should not have had m = tan α.
9. The Circle.—It is easy to write down the equation of a given circle. Let (h, k) be its given centre C, and ρ the numerical measure of its given radius. Take P (x, y) any point on its circumference, and construct the triangle CRP, in fig. 50 as above. The fact that this is right-angled tells us that
and this at once gives the equation
A point not upon the circumference of the particular circle is at some distance from (h, k) different from ρ, and satisfies an equation inconsistent with this one; which accordingly represents the circumference, or, as we say, the circle.
The equation is of the form
Conversely every equation of this form represents a circle: we have only to take −A, −B, A2 + B2 − C for h, k, ρ2 respectively, to obtain its centre and radius. But this statement must appear too unrestricted. Ought we not to require A2 + B2 − C to be positive? Certainly, if by circle we are only to mean the visible round circumference of the geometrical definition. Yet, analytically, we contemplate altogether imaginary circles, for which ρ2 is negative, and circles, for which ρ = 0, with all their reality condensed into their centres. Even when ρ2 is positive, so that a visible round circumference exists, we do not regard this as constituting the whole of the circle. Giving to x any value whatever in (x − h)2 + (y − k)2 = ρ2, we obtain two values of y, real, coincident or imaginary, each of which goes with the abscissa x as the ordinate of a point, real or imaginary, on what is represented by the equation of the circle.
The doctrine of the imaginary on a circle, and in geometry generally, is of purely algebraical inception; but it has been in its entirety accepted by modern pure geometers, and signal success has attended the efforts of those who, like K. G. C. von Staudt, have striven to base its conclusions on principles not at all algebraical in form, though of course cognate to those adopted in introducing the imaginary into algebra.
A circle with its centre at the origin has an equation x2 + y2 = ρ2.
In oblique coordinates the general equation of a circle is x2 + 2xy cos ω + y2 + 2Ax + 2By + C = 0.
10. The conic sections are the next simplest loci; and it will be seen later that they are the loci represented by equations of the second degree. Circles are particular cases of conic sections; and
