will, by a similar substitution, become

x cos (0'+0') + y sin (0′+0′′) = r cos≥ (0′ – 0′′),

0' and 0" being the angles which radii drawn to the extremities of the chord make with the axis of x.

This equation might also have been obtained directly from the general equation of a right line (Art. 25),

x cosa + y sina = p,

for the angle which the perpendicular on the chord makes with the axis is plainly half the sum of the angles made with the axis by radii to its extremities; and the perpendicular on the chord

= r cos (0′ – 0'').

Ex. 1. To find the co-ordinates of the intersection of tangents at two given points on the circle. The tangents being

Ex. 2. To find the locus of the intersection of tangents at the extremities of a chord whose length is constant.

it reduces to cos (0′ – 0′′)= const., or 0′ — 0′′ = const. If the given length of the chord be 2r sind, then ′-0′′ 28. The co-ordinates then found in the last example fulfil the condition

(x2 + y2) cos2d = r2.

Ex. 3. What is the locus of a point where a chord of a constant length is cut in a given ratio?

Writing down (Art. 7) the co-ordinates of the point where the chord is cut in a given ratio, it will be found that they satisfy the condition x2 + y2 : = const.

Ex. 4. The diagonals of a hexagon circumscribing a circle meet in a point. Let the angles made with the axis by radii to the points of contact be 2a, 28, 2y, 28, 28, 29; then the equation of the line joining the intersection of the tangents at 2a, 26, to that of the tangents at 28, 2ɛ, will be

1

1

sin (a − ô) { x cos (a + d) + y sin (a + d)

− r cos (a - d)} + · { x cos (ẞ + ε) + y sin (ẞ+ ε) — r cos (Be)} = 0; which, sin (3-ε) when added to the other two equations of like form, vanishes identically.

101. We have seen that the tangent to any circle x2 + y2 = r2 has an equation of the form

x cos 0 + y sin 0 =r;

and it would appear, in like manner, that the equation to the tangent to (a)+(y-b) = r2 may be written.

conversely, then, if the equation of any right line contain an indeterminate in the form

that right line will touch the circle

( x − a)2 + (y − b)2 = r2.

Ex. 1. If a chord of a constant length be inscribed in a circle, it will always touch another circle. For, in the equation of the chord

x cos (0′ + 0′′) + y sin § (0′ + 0′′) = r cos § (0′ – 0′′)

O" is known, and 0′+ 0′′ indeterminate; the chord, therefore,

Ex. 2. Given any number of points, if a right line be such that m' times the perpendicular on it from the first point, + m" times the perpendicular from the second, + &c., be constant, the line will always touch a circle.

This only differs from the question, p. 48, in that the sum, in place of being = 0, is constant. Adopting then the notation of that Article, instead of the equation there found,

{xΣ(m) — Σ(mx')} cosa + {yΣ(m) – Σ(my')} sina :

we have only to write

{xΣm - (mx')} cosa + {yΣ(m) - (my)} sin a

Hence this line must always touch the circle

= const.

whose centre is the centre of mean position of the given points.

102. We shall conclude this Chapter with some examples of

the use of polar co-ordinates.

Ex. 1. If through a fixed point any chord of a circle be drawn, the rectangle under

its segments will be constant (Euclid, III. 35, 36).

Take the fixed point for the pole, and the polar equation is (Art. 93)

p2 - 2pd cos + d2 − r2 = 0 ;

the roots of which are evidently OP, OP', the values of the radius vector answering to any given value of 0 or POC.

Now, by the theory of equations, OP. OP, the product of these roots will = d2 — r2, a quantity independent of 0, and therefore constant, whatever be the direction in which the line OP is drawn. If the point O be outside the circle, it is plain that d2 - r2 must the square of the tangent.

be

=

Ex. 2. If through a fixed point O any chord of a circle be drawn, and OQ taken an arithmetic mean between the segments OP, OP'; to find the locus of Q.

=

We have OP+ OP', or the sum of the roots of the quadratic in the last example 2d cos 0; but OP + OP' = 20Q, therefore,

The question in this example might have been otherwise stated: "To find the locus of the middle points of chords which all pass through a fixed point."

Ex. 3. If the line OQ had been taken an harmonic mean between OP and OP', to find the locus of Q.

This is the equation of a right line (Art. 44) perpendicular to OC, and at a distance

We can, in like manner, solve this and similar questions when the equation is given in the form

Ax2+ Ay2+ Dx + Ey + F = 0,

for, transforming to polar co-ordinates, the equation becomes

and, proceeding precisely as in this example, we find, for the locus of harmonic means,

and, returning to rectangular co-ordinates, the equation of the locus is

Dx + Ey+2F = 0,

the same as the equation of the polar obtained already (Art. 87).

Ex. 4. Given a point and a right line; if OQ be taken inversely as OP, the radius vector to the right line, find the locus of Q.

Ex. 5. Given vertex and vertical angle of a triangle and rectangle under sides; if one base angle describe a right line or a circle, find locus described by the other base angle.

Take the vertex for pole; let the lengths of the sides be p and p', and the angles they make with the axis 0 and 0', then we have pp' = k2 and 0 0' = C.

The student must write down the polar equation of the locus which one base angle is said to describe; this will give him a relation between and 0; then, writing for k2

P1 and for 0, CO', he will find a relation between p' and ', which will be the

-

polar equation of the locus described by the other base angle.

This example might be solved in like manner, if the ratio of the sides, instead of their rectangle, had been given.

Ex. 6. Through the intersection of two circles a right line is drawn; find the locus of the middle point of the portion intercepted between the circles.

The equations of the circles will be of the form,

p = 2r cos (0 - a); p = 2r' cos (0 - a');

and the equation of the locus will be

which also represents a circle.

= r cos(0 − a) + r' cos (0 — a') ;

Ex. 7. If through any point O, on the circumference of a circle, any three chords be drawn, and on each, as diameter, a circle be described, these three circles (which, of course, all pass through O) will intersect in three other points, which lie in one right line. (See Cambr. Math. Jour., I. 169.)

Take the fixed point O for pole, then if d be the diameter of the original circle, its polar equation will be (Art. 93)

ρ

= d cos 0.

In like manner, if the diameter of one of the other circles make an angle a with the fixed axis, its length will be d cosa, and the equation of this circle will be

=

To find the polar co-ordinates of the point of intersection of these two, we should seek what value of would render

cosa. cos (0 - a) = cos ẞ.cos (0 - ß),

and it is easy to find that 0 must = a + ẞ, and the corresponding value of p = d cos a cos B. Similarly, the polar co-ordinates of the intersection of the first and third circles are

Now, to find the polar equation of the line joining these two points, take the general equation of a right line, p cos (k − 0) = p (Art. 44), and substitute in it successively these values of 0 and p, and we shall get two equations to determine p and k. We shall get

Hence

k = a + B + Y, and Р =

p = d cos a cosẞ cos (k - a + B) = d cosa cos y cos (k
d cos u cosẞ cosy.

a +

The symmetry of these values shows that it is the same right line which joins the intersections of the first and second, and of the second and third circles, and, therefore, that the three points are in a right line.

* CHAPTER VIII.

APPLICATION OF ABRIDGED NOTATION TO THE EQUATION OF THE CIRCLE.

103. If we have an equation of the second degree expressed in the abridged notation explained in Chap. IV., and if we desire to know whether it represents a circle, we have only to transform to x and y co-ordinates, by substituting for each abbreviation (a) its equivalent (a cosa + y sin a − p); and then to examine whether the coefficient of xy in the transformed equation vanishes, and whether the coefficients of a2 and of y' are equal. This is sufficiently illustrated in the examples which follow.

When will the locus of a point be a circle if the product of perpendiculars from it on two opposite sides of a quadrilateral be in a given ratio to the product of perpendiculars from it on the other two sides?

=

Let a = 0, B = 0, y = 0, 80 be the equations of the four sides of the quadrilateral, then the equation of the locus is at once written down ay kẞd, which represents a curve of the second degree passing through the angles of the quadrilateral; since it is satisfied by any of the four suppositions,

=

a = 0, ẞ = 0; a = 0, d = 0; ß = 0, y = 0; ́ ß = 0, 8 = 0.

Now, in order to ascertain whether this equation represents a circle, write it at full length

t