This is the converse of the preceding proposition, and we may prove it as follows:

Let y=mx+b be the equation to the straight line and (x11) any point in it. Draw tangents to the circle from this point; the equation to the chord of contact is xx1+yy1 = c2.

Since the line y=mx+b_passes through (x11) we may write Y1 =mx1+b. Substituting for y1 in xx1+yy1 =c2 we have

xx1+yx1m+yb=c2, or (x+my)x1+yb - c2=0...(1).

Here we must have r+my=0, and also yb - c2=0, whatever be the value of x1; that is, the straight line represented by (1) passes through the points whose co-ordinates are y=! x= -m. values

2

b'

b'

which are found from the simultaneous equations x+my=0 and yb-c2 = 0.

58. To find the equation to the normal at any point (x1y1) of the circle x2+y2=c2.

The usual way is to deduce it from the equation to the tangent. But it may be obtained directly from the figure in this way: The form of equation to a straight line through the origin, which in this case is the centre of the circle, is y = mx. But it is evident from

the figure that m= 1. Hence the equation required is y=x; for

x1

21 the line it represents passes through the point (x11) and is, by the nature of the tangent, perpendicular to the tangent at that point.

59. To show that the locus of the vertices of all right-angled triangles on a diameter of the circle (x2+y2=c2) as base is a semicircle. Construct the figure, and it will be evident that we may take the equations

1

(x+c)... (2)

μ

as convenient representations of the two straight lines which meet at the vertex of any right-angled triangle whose base is a diameter of the above circle.

For these equations obviously represent all straight lines which fulfil the following three conditions :—

(a) They are at right angles to each other; for the condition of their perpendicularity is that μ= where μ is an arbitrary

constant.

1

μ

(b) They pass through the extremities of the diameter, which we may take as the axis of x. For (1) may be written y-μ (x − c)=0, thus representing a line passing through the intersection of the lines y=0, x-c=0, a point at that extremity of the diameter which is on the positive part of the axis of x.

Similarly, (2) passes through the point y=9, x= −c, the other extremity of the diameter.

(c) They intersect in the given circle; for, multiplying (1) and (2) together, we eliminate μ, and obtain the equation to the circle.

Now, since (1) and (2) represent all possible lines fulfilling the above conditions, we may conclude that the locus of the vertices of all right-angled triangles, &c.

Thus the equation x2+y2 = c2 suggests to us the above well-known property of the circle. For the equation may be considered as representing the two straight lines y=μ (x-c), y= (See Article 32.)

The Use of the more General Equation.

(x + c). μ

60. Before we pass to the propositions which refer to oblique and polar co-ordinates, we would strongly impress upon the candidate the necessity of solving most of the above problems, especially those which refer to the tangent and normal, by using the more general form of equation to the circle, (x − a)2+(y - b)2 = c2.

Of course, he may obtain the results required in this case by first using the simple form x2+ y2=c2, and then, by Transformation of Co-ordinates, transferring the origin to the point (ab). Here a and b are the co-ordinates of the old origin referred to the new, just as -α and -b would be the co-ordinates of the new origin referred to the old.

For example, in the equation to the tangent write

x-a for x, x1-a for x1, y-b for y, and y1-b for y1.

The equation then becomes

(x − α) (x 1 − α) + (y − b) (y1 − b) =c2,

which is the equation to the tangent to the circle (x − a)2 + (y − b)o =c2 at any point (x11).

Similarly the equation to the normal to this circle may be found

to be y-b:

=

(x-a); or, rather y-y1 =

yi-b x1 -a

y1-b (x-x1); for X1 - a the normal is considered as a straight line passing through the point (x171) rather than (ab), although it really passes through both points in the case of the circle.

But the student should exercise himself in the more usual way of deducing these equations to the tangent and normal on the principle of the general definitions of these lines. "If two points be taken on a curve and a secant drawn through them; then, if the first point remains fixed, and the second moves on the curve up to the first, the secant in its limiting position is called the tangent to the curve at the first point." "The normal at any point of a curve is a straight line drawn through that point at right angles to the tangent at that point."

Similarly, the student should test his comprehension of the subject by investigating the equations to these lines when the equation to the circle is given in the particular cases

x2+ y2 - 2ax - 2by=0, x2 + y2 − 2ax = 0, x2 + y2 - 2by=0,

and also in those of (6), Article 37.

The tangents in the above three cases are respectively :

xx1+Yy1-α(x − x1) — b(y−y1) =0,

He should, moreover, endeavour to write down the form of equation to the tangent when the circle is given in the general form x2 + y2+Dx + Ey + F= 0.

He may also, for the sake of exercise, inquire what the equation to the tangent line in all the above cases will be when expressed in terms of the tangent of the angle which the straight line makes with the axis of x. (See Article 47.)

Polar Equation.

61. To find the equation to the circle referred to polar co-ordinates. When r1, 01 are the co-ordinates of the centre, and r, @ the co-ordinates of any point in the circumference, the equation to the circle whose radius is c is r2+r2 - 2r1 rcos(0 - 01) = c2.

Compare this with Article 1, and observe that the formula simply indicates that the distance between the points (r11) and (re) is constant and equal to the radius c. In this article (re) is a movable point; r, change in value as the point moves, and hence are called the variables of the equation.

NOTES. (1) Write the equation as a quadratic in r, viz.:— p2 - 2r1 cos(0 – 01) r + r2 − c2 = 0.

The two values of r are the distances of the pole from the points where the radius vector cuts the circle. Suppose a and B are the two roots of the equation corresponding to any value of 0; then, by Algebra, aẞ=r2-c2, a quantity which is the same for the different values of 0. This shows that if from any point a straight line be drawn to cut a circle, the rectangle contained by its segments is constant.

If the line is a tangent to the circle, we shall have a =ẞ, or the two values of r will be equal, Then a2=r2c2, or the square of the distance between the pole and the point of contact is equal to the rectangle contained by the segments of any straight line drawn from the pole to cut the circle.

(2) If the pole be within the circle, then we have r2<r2; hence a, ẞ have opposite signs, or are drawn in opposite directions.

(3) If the centre of the circle become the pole, then r1 =0, and we have for the equation in this case the simple form r=c.

(4) If the pole be on the circumference, and the diameter through the pole make an angle 0, with the initial line, the equation in this case is r=2c cos(0-01); for then r1 = c.

(5) If the diameter be the initial line and one extremity of it the pole, then r1 = c, and 01=0. Hence the equation in this case becomes r = 2c cose.

(6) If the initial line be a diameter, then 01 =0, and the equation becomes r2+2 − 2r1r cos✪ = c2.

The student should exercise his mind and train his eye by considering what the polar equation becomes when the circle is placed in other positions, such as when (1) either the initial line or the radius vector is a tangent to the circle; or (2) both of them are tangents at the same time; (3) when the initial line cuts the circle.

All the above cases should be verified by means of appropriate geometrical figures, from which the student may obtain the various forms of equations which correspond to the particular conditions.

(7) Since x=r cose and y=r sine are the equations which connect the rectangular and polar co-ordinates of a point, we may deduce the polar equation to the circle from the equation referred to rectangular axes by substituting the above values of x and y in the equation (x − a)2 + (y − b)2 = c2 and observing that r12=a2+b3, a=r1 cose1, and b=r1 sino.

(8) The general form of the polar equation to the circle is

r+Ar cose +B r sin0+C=0,

where the variables are r and 0, and the constant quantities are A, B, C, any one or more of which may under particular conditions be equal to zero.

62. To express the perpendicular (p) from the origin on the tangent at any point of the circle in terms of the radius-vector of that point. We shall get "1 2 12= r2+ c2 - 2cp, or r12= r2+c2+2cp according as the tangent at the point falls on the same side of the pole and the centre or between the pole and the centre.

1

63. Show that the polar equation of the locus of the middle point of the chord which the circle cuts off from the radius-vector is a circle.

Taking the form in (6), Article 61,

2

72-2r,cose r+r12 - c2=0,

by Algebra we know that the sum of the two values of r is 2r,cose. Now the polar co-ordinates of the middle point of the chord above referred to are (2r,cose) and 0, or r1 cose and 0. Hence we have r=rcose as the locus of the middle point. But this equation by (5), Article 61, represents a circle of which the diameter

In concluding our outline of this subject we may observe that we have given some propositions which are not likely to be required of the Pass candidate, especially those which relate to oblique co-ordinates; but the mode of reasoning may suggest many valuable points to the careful reader, and it is well for every candidate to see the wide application of which the brief statement in the Regulations is capable. It now only remains for us to give a brief account of the text-books where he may find all the help he will really require.

TEXT-BOOKS IN PLANE CO-ORDINATE GEOMETRY.

Perhaps the simplest for a mere beginner is Hunter's Easy Introduction to the Higher Treatises on the Conic Sections (3s. 6d., Key 28., Longmans). The author having "ascertained the nature of the difficulties that perplex and discourage many readers" of the Higher Treatises-Salmon's, Hymers's, Puckle's, and Todhunter's"has endeavoured in this publication to provide a first course of lessons and exercises which, he thinks, cannot fail of qualifying the young student for reading with intelligence and profit the masterly works referred to. Todhunter's book is especially recommended as the most appropriate sequel to this Introduction." Mr. Hunter's little work has certainly the merit of clearness, simplicity, and of ample illustration in the form of worked examples; and many a sharp student who had well mastered it might, before the recent change in the Regulations, have answered many of the ExaminationPapers in this subject by its aid alone. But the case is different now, when the equations are no longer referred to rectangular co-ordinates alone. Mr. Hunter always supposes the axes to be rectangular; he has nothing to say on polar co-ordinates; and therefore his work is. not a complete introduction to what is required. It may, however, be used with great advantage by the pupil who has never read in the subject before and wishes to make a rapid run through its simpler elements before attempting to master a more comprehensive treatise. Hann's Rudimentary Treatise on Analytical Geometry and Conic Sections (2s., Lockwood and Co.) carries the pupil farther than the previous work. It includes oblique axes of reference as well as rectangular, but does not deal with polar co-ordinates. The new edition is rewritten and enlarged by Professor Young. It contains Examples for Exercise," and a serviceable and judicious selection of worked examples; moreover it enters into a more detailed explanation and discussion of certain difficulties and important principles than elementary treatises usually do. Here and there, too, the properties of the straight line and the circle are deduced from their equations. It is a work which will be very helpful to many a candidate who will refer to it for suggestive views of much that is left unexplained in other text-books, to any of which it is a fitting companion.