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CHAPTER VI.

THE CIRCLE.

79. BEFORE proceeding to the discussion of the general equation of the second degree, it seems desirable that we should show in the simple case of the circle, how all the properties of a curve may be deduced from its equation, without assuming any previous acquaintance with the geometrical theory.

The equation, to rectangular axes, of the circle whose centre is the point (a) and radius is r, has already (Art. 17) been found to be

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Two particular cases of this equation deserve attention, as occurring frequently in practice. Let the centre be the origin, then a=0, B=0, and the equation is

x2 + y2 = r2.

Let the axis of x be a diameter, and the axis of y a perpendicular at its extremity, then a=r, ß=0, and the equation becomes

x2 + y2 = 2rx.

80. It will be observed that the equation of the circle, to rectangular axes, does not contain the term xy, and that the coefficients of x2 and y2 are equal. The general equation therefore ax2+2hxy+by2+2gx + 2fy + c = 0,

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cannot represent a circle, unless we have h=0, and a=b. Any equation of the second degree which fulfils these two conditions may be reduced to the form (x − a)2 + (y − B)2 = r2, by a process corresponding to that used in the solution of quadratic equations. If the common coefficient of x and y be not already unity, by division make it so; then having put the terms containing x and y on the left-hand side of the equation, and the constant term on the right, complete the squares by adding to both sides the sum of the squares of half the coefficients of x and y.

Ex. Reduce to the form (x - a)2 + (y − ẞ)2 = r2, the equations

x2 + y2 - 2x-4y = 20; 3x2 + 3y2 - 5x - 7y + 1 = 0.

Ans. (x-1)2 + (y − 2)2 = 25; (x − 3)2 + (y − 3)2 = 3; and the co-ordinates of the centre and the radius are (1, 2) and 5 in the first case; (, ) and √(62) in the second.

If we treat in like manner the equation

a(x2+ y2)+2gx+2fy+c=0,

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and the co-ordinates of the centre are -9 -f

and the radius

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a

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If g2 +ƒ* is less than ac, the radius of the circle is imaginary, and the equation being equivalent to (x − a)2 + (y − ẞ)2 + r2 = 0, cannot be satisfied by any real values of x and y.

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If g2+f*=ac, the radius is nothing, and the equation being equivalent to (x − a)2 + (y − ẞ)2 = 0, can be satisfied by no coordinates save those of the point (aß). In this case then the equation used to be called the equation of that point, but for the reason stated (Art. 73) we prefer to call it the equation of an infinitely small circle having that point for centre. We have seen (Art. 73) that it may also be considered as the equation of the two imaginary lines (x− a) + (y — B) √(− 1) passing through the point (a). So in like manner the equation x2+y=0 may be regarded as the equation of an infinitely small circle having the origin for centre, or else of the two imaginary lines x±y√(−1).

81. The equation of the circle to oblique axes is not often used. It is found by expressing (Art. 5), that the distance of any point from the centre is equal to the radius; and is

As ofcentre

(x − a)2 + 2 (x − a) (y − B) cos w + (y − B)” = r2.

If we compare this with the general equation, we see that the latter cannot represent a circle unless a = b, and ha cosa. When these conditions are fulfilled, we find by comparison of coefficients that the co-ordinates of the centre and the radius are given by the equations

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Since a, ẞ are determined from the first two equations which do not contain c, we learn that two circles will be concentric if their equations differ only in the constant term.

For then the

Again, if c=0, the origin is on the curve. equation is satisfied by the co-ordinates of the origin x = 0, y = 0. The same argument proves that if an equation of any degree want the absolute term, the curve represented passes through the origin.

82. To find the co-ordinates of the points in which a given right line x cosa + y sina=p, meets a given circle x2 + y2 = r2. Equating to each other the values of y found from the two equations, we get for determining x, the equation

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(The reader may satisfy himself, by substituting these values in the given equations, that the in the value of y corresponds to the in the value of x, and vice versa.)

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Since we obtained a quadratic to determine x, and since every quadratic has two roots, real or imaginary; we must, in order to make our language conform to the language of algebra, assert that every line meets a circle in two points, real or imaginary. Thus, when p is greater than r, that is to say, when the distance of the line from the centre is greater than the radius, the line, geometrically considered, does not meet the circle; yet we have seen that analysis furnishes definite imaginary values for the co-ordinates of intersection. Instead then of saying that the line meets the circle in no points, we shall say that it meets it in two imaginary points, just as we do not say that the corresponding quadratic has no roots, but that it has two imaginary roots. By an imaginary point we mean nothing more than a point, one or both of whose co-ordinates are imaginary. It is a purely analytical conception, which we do not attempt to represent geometrically; just as when we find imaginary values for roots of an equation, we do not try to attach an arithmetical

meaning to our result. points is necessary to preserve generality in our reasonings, for we shall presently meet with many cases in which the line joining two imaginary points is real, and enjoys all the geometrical properties of the corresponding line in the case where the points are real.

And attention to these imaginary

83. When p=r, it is evident geometrically that the line touches the circle, and our analysis points to the same conclusion, since the two values of x in this case become equal, as do likewise the two values of y. Consequently the points answering to these two values, which are in general different, will in this case coincide. We shall therefore, not say that the tangent meets the circle in only one point, but rather that it meets it in two coincident points; just as we do not say that the corresponding quadratic has only one root, but rather that it has two equal roots. And in general we define the tangent to any curve as the line joining two indefinitely near points on that curve.

We can in like manner find a quadratic to determine the points where the line Ax+By+C meets a circle given by the general equation. When this quadratic has equal roots, the line is a tangent.

Ex. 1. Find the co-ordinates of intersection of x2 + y2 = 65; 3x + y = 25.

Ans. (7, 4) and (8, 1).

Ex. 2. Find intersections of (x —c)2 + (y − 2c)2 = 25c2; 4x + 3y = 35c.
Ans. The line touches at the point (5c, 5c).
When b2 = r2 (1 + m2).

Ex. 3. When will y = mx + b touch x2 + y2 = r2? Ans.
Ex. 4. When will a line through the origin, y = mx, touch
a (x2 + 2xy cos w + y2) + 2gx + 2fy + c?

The points of meeting are given by the equation

a (1 + 2m cos w + m2) x2 + 2 (g+fm) x + c = 0,

which will have equal roots when

(g+fm)2 = ac (1 + 2m cos w + m2).

We have thus a quadratic for determining m.

Ex. 5. Find the tangents from the origin to x2 + y2 — 6x − 2y + 8 = 0.

Ans. x - y = 0, x + 7y = 0.

84. When seeking to determine the position of a circle represented by a given equation, it is often as convenient to do so by finding the intercepts which it makes on the axes, as by finding its centre and radius. For a circle is known when

three points on it are known; the determination, therefore, of the four points where the circle meets the axes serves completely to fix its position. By making alternately y=0, x=0 in the general equation of the circle, we find that the points in which it meets the axes are determined by the quadratics

ax2+2gx + c = 0, ay2+2fy+c=
= 0.

The axis of x will be a tangent when the first quadratic has equal roots, that is, when g2= ac; and the axis of y when ƒ*= ac. Conversely, if it be required to find the equation of a circle making intercepts λ, X' on the axis of x, we may take a = 1, and we must have 2g = − (λ+λ'), c = XX'. If it make intercepts c=λa'. μ, μ' on the axis of y, we must have 2f=-(μ+μ'), c=μμ'. Thus we see that we must have Xλ' =μμ' (Euc. III. 36).

Ex. 1. Find the points where the axes are cut by x2 + y2 - 5x − 7y+ 6 = 0. Ans. x3, x = = 2; y = 6, y = 1. Ex. 2. What is the equation of the circle which touches the axes at distances from the origin =a? Ans. x2+ y2 - 2ax - 2ay + a2 = 0.

Ex. 3. Find the equation of a circle, the axes being a tangent, and any line through the point of contact. Here we have λ, λ', μ all = 0; and it is easy to see from the figure that μ' = 2r sin w, the equation therefore is

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85. To find the equation of the tangent at the point x'y' to a given circle.

The tangent having been defined (Art. 83) as the line joining two indefinitely near points on the curve, its equation will be found by first forming the equation of the line joining any two points (x'y', x"y") on the curve, and then making x'=x" and y=y" in that equation.

To apply this to the circle: first, let the centre be the origin, and, therefore, the equation of the circle x2 + y2 = r2.

The equation of the line joining any two points (x'y') and (x"y") is (Art. 29)

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now if we were to make in this equation y' =y" and x' =x", the right-hand member would become indeterminate. The cause of this is, that we have not yet introduced the condition, that the two points (x'y', x'y'") are on the circle. By the help of this condition we shall be able to write the equation in a form which

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