Online Library → S. L. (Sidney Luxton) Loney → The elements of coordinate geometry → online text (page 8 of 26)

Font size

sin^ lo sin^ w' '

ab-h^ db'-K''

and . o =^ â€” . o / ,

sm" (i) sm^ o)

0) a7id oi' being the angles between the original and final indrs

of axes.

Let the coordinates of any point P, referred to the

original axes, be x and y and, referred to the final axes, let

them be x and y .

By Art. 20 the square of the distance between P and

the origin is o^ + 2xy cos a> + 2/^, referred to the original axes,

and x'^ + 2xy cos w' + y"^^ referred to the final axes.

We therefore always have

X- + Ixy cos w + 2/^ = a;'^ + ^xy cos m -vy''^ (1).

Also, by supposition, we have

ax"" + 2hxy + by"" = ax'- + 2h'xy' + b'lj^ (2).

Multiplying (1) by \ and adding it to (2), we therefore have

x^ (Â« + X) + 2xy (h + X cos to) +y^ (b + X)

= x"" (a' + X) + 2x'y' {h' + X cos o>') + y" {b' + X) . . .(3).

If then any value of X makes the left-hand side of (3) a

perfect square, the same value must make the right-hand

side also a perfect square.

But the values of X which make the left-hand a perfect

square are given by the condition

(h + X cos (o)2 -(a + X) {b + X),

EXAMPLES. 117

i.e. by

\2 (1 - cos^ w) + X{a + h-2h cos oi) + ab- h^ = 0,

^â€ž . a + h â€” 2hcos(ji ab â€” h^^ ,,,

i.e. by X^ + X r^ + -r-^ â€” = (4).

In a similar manner the values of A, which make the

right-hand side of (3) a perfect square are given by the

equation

^a' + b'-2h'cosoy' a'b'-h'^ ^

X' + X r^-^ + . ^ , =0 (5).

sm^ to sm'^ci) ^ '

Since the values of \ given by equation (4) are the same

as the values of X given by (5), the two equations (4) and

(5) must be the same.

Hence we have

a + b â€” 2h cos cu a' + b' â€” 2h' cos co'

and

sin^ (o sin^ co'

ab-h2 a'b'-h'2

sin2 0) sin2 cd'

EXAMPLES. XVI.

1. The equation to a straight line referred to axes inclined at 30Â°

to one another is y = 2x + l. Find its equation referred to axes

inclined at 45Â°, the origin and axis of x being unchanged.

2. Transform the equation 2x'^ + 3 sjSxy + Sy^ = 2 from axes

inclined at 30Â° to rectangular axes, the axis of x remaining

unchanged.

3. Transform the equation x^ + xy + y^ = 8 from axes inclined at

60Â° to axes bisecting the angles between the original axes.

4. Transform the equation y'^-\-4cy cot a - 4cc=0 from rectangular

axes to oblique axes meeting at an angle a, the axis of x being kept

the same.

5. If aj and y be the coordinates of a point referred to a system of

obUque axes, and x' and y' be its coordinates referred to another

system of oblique axes with the same origin, and if the formulae of

transformation be

x=mx' + ny' and y â€” m'x'-\-n'y\

., . m^ + m'^-1 mm'

prove that Â» , ,. ., = â€” 7- .

CHAPTER VIII.

THE CIRCLE.

138. Def. A circle is the locus of a point which

moves so that its distance from a fixed point, called the

centre, is equal to a given distance,

called the radius of the circle.

The given distance is

139. To find the equation to a circle, the axes of coordi-

nates being two straight lines through its centre at right

angles.

Let be the centre of the circle and let a be its radius.

Let OX and OF be the axes of

coordinates.

Let P be any point on the circum-

ference of the circle, and let its coordi-

nates be X and y.

Draw PJf perpendicular to OX and

join OP.

Then (Euc. I. 47)

OM^ + MP' = a',

i.e. x2 + y2 = a2.

This being the relation which holds between the coordi-

nates of any point on the circumference is, by Art. 42, the

required equation.

140. To find the equation to a circle referred to any

rectangular axes.

THE CIRCLE.

119

Let OX and (9 F be the two rectangular axes.

Let C be the centre of the

circle and a its radius.

Take any point P on the

circumference and draw per-

pendiculars CM and FN upon

OX ; let F be the point (x, y).

Draw GL perpendicular to

NF.

Let the coordinates of G be

h and h ; these are supposed to be known.

We have GL = MN= ON- OM=x- h,

and LF = NF-NL = NF-MG^y-h.

Hence, since GL^ + LF^ = GF\

we have (x-h)2+ (y-k)2 = a2 (1).

This is the required equation.

Ex. The equation to the circle, whose centre is the point ( - 3, 4)

and whose radius is 7, is

i.e. a;2 + 2/2 + 6a:-8?/ = 24.

141. Some particular cases of the preceding article may be

noticed :

(a) Let the origin be on the circle so that, in this case,

i.e. h^ + Jc'^=a^.

The equation (1) then becomes

i.e. x'^ + y^-2hx-2ky = 0.

(jS) Let the origin be not on the curve, but let the centre lie on

the axis of x. In this case k = 0, and the equation becomes

{x - lif â– \-y^ = a^.

(7) Let the origin be on the curve and let the axis of re be a

diameter. We now have fe = and a = h, so that the equation becomes

x^ + t/-2hx = 0.

(8) By taking at C, and thus making both h and k Tiero, we

have the case of Art. 139.

120 COORDINATE GEOMETRY.

(e) The circle will touch the axis of x if MG be equal to the

radius, i.e. if k = a.

The equation to a circle touching the axis of x is therefore

x'^ + 7f-2hx-2ky + h'^ = Q.

Similarly, one touching the axis of y is

x^ + y^-2hx-2ky + k^ = 0.

142. To prove that the equation

a^ + 2/'+2^a;+2/2/ + c = (1),

always represents a circle for all values of g,f and c, and to

find its centre and radius. [The axes are assumed to be

rectangular.]

This equation may be written

{a? + 2gx + cf) + {f + 2/2/ +/^) = cf +f' - c,

i.e. {X + gf 4- {y +ff = {JfTp^cf.

Comparing this with the equation (1) of Art. 140, we

see that the equations are the same if

h = -g, h^-f and a = Jg"" -hf^-c.

Hence (1) represents a circle whose centre is the point

(â€” ^, â€”f), and whose radius is Jg^-^f^ â€” c.

If g^ +f' > c, the radius of this circle is real.

If g^ +f'^ = c, the radius vanishes, i. e. the circle becomes

a point coinciding with the point (â€” ^, â€”f). Such a circle

is called a point-circle.

If g^ +f^ < c, the radius of the circle is imaginary. In

this case the equation does not represent any real geo-

metrical locus. It is better not to say that the circle does

not exist, but to say that it is a circle with a real centre

and an imaginary radius.

Ex. 1. The equation x"^ + ij^ + 4^x - 6y = can be written in the

form

{x + 2f + iy-Bf = lS = {JlS)\

and therefore represents a circle whose centre is the point ( - 2, 3) and

whose radius is >,yi3.

GENERAL EQUATION TO A CIRCLE. 121

Ex. 2. The equation 45a;2 + 45?/2 - 60x- + SQy + 19 = is equivalent

to

i.e. (Â«'-|)^ + (y + F=f + ^*5-H=/A,

and therefore represents a circle whose centre is the point (f , -|) and

whose radius is â€”=- .

15

143. Condition that the general equation of the second

degree may represent a circle.

The equation (1) of the preceding article, multiplied by

any arbitrary constant, is a particular case of the general

equation of the second degree (Art. 114) in which there is

no term containing xy and in which the coefficients of x^

and y^ are equal.

The general equation of the second degree in rectangular

coordinates therefore represents a circle if the coefficients

of x^ and y^ be the same and if the coefficient of xy

be zero.

144. The equation (1) of Art. 142 is called the

general equation of a circle^ since it can, by a proper

choice of g, f and c, be made to represent any circle.

The three constants g, f, and c in the general equation

correspond to the geometrical fact that a circle can be found

to satisfy three independent geometrical conditions and no

more. Thus a circle is determined when three points on it

are given, or when it is required to touch three straight

lines.

145. To find the equation to the circle which is described on the

line joining the points {x-^ , y-^) and (x^ , 2/2) ^^ diameter.

Let A be the point (a;^, y-^) and B be the point {x^, 2/2) Â» ^^^ ^^^ ^^^

coordinates of any point P on the circle be h and k.

The equation to AP is (Art. 62)

y-yi=j^i^-^i) (1).

and the equation to BP is

2/ -2/2=^' (^-^2) (2).

But, since APB is a semicircle, the angle APB is a right angle,

and hence the straight Unes (1) and (2) are at right angles.

122

COOKDINATE GEOMETRY.

Hence, by Art. 69, we have

^-Vi _ h^ljh^ _ 1

i.e. (h-x-^){h-x^) + {k-yj}{Jc-y2) = 0.

But this is the condition that the point {h, k) may lie on the curve

whose equation is

{X - x^) {x - x^) + {y- y^) {y - y^) = 0.

This therefore is the required equation.

146. Intercepts made on the axes by the circle whose equation is

ax^ + ay^ + 2gx + 2fy + c = (1).

The abscissae of the points where the circle (1) meets the axis of x,

i.e. y = 0, are given by the equation

ax^ + 2gx + c = (2).

The roots of this equation being Xj^ and x^ ,

we have

^ + â€¢^2

29

a 'â–

and

(Art. 2.

Hence

A-^A^ = x^ -x-^= J{x-^-\-x^^ - 4x-^i

V a2

= 2

a a

Again, the roots of the equation (2) are both imaginary if g^<ac.

In this case the circle does not meet the axis of x in real points, i.e.

geometrically it does not meet the axis of x at all.

The circle will touch the axis of x if the intercept A-^A^, be just

zero, i.e. ii g^ = ac.

It will meet the axis of x in two points lying on opposite sides of

the origin if the two roots of the equation (2) are of opposite signs,

i. e. if c be negative.

147. Ex.1. Find the equation to the circle which passes through

the points (1, 0), (0, - 6), and (3, 4).

Let the equation to the circle be

x^ + y^ + 2gx + 2fy + c = (1).

Since the three points, whose coordinates are given, satisfy this

equation, we have

l + 2^ + c = (2),

36-12/+c=0 (3),

and 25 + 6^ + 8/+c = (4).

EXAMPLES. 123

Subtracting (2) from (3) and (3) from (4), we have

2^ + 12/= 35,

and 65f + 20/=ll.

Hence /=*/- and g=z -iJ-.

Equation (2) then gives c = ^-.

Substituting these values in (1) the required equation is

4a;2 + 4i/2 - 142a; + 47a; + 138 = 0.

Ex. 2. Find the equation to the circle which touches the axis ofy

at a distance + 4 from the origin and cuts off an intercept 6 from the

axis of X.

Any circle is x^ + y'^ + 2gx + 2fy + c = 0.

This meets the axis of y in points given by

y^ + 2fy + c = 0.

The roots of this equation must be equal and each equal to 4, so

that it must be equivalent to {y - 4)^ = 0.

Hence 2/= -8, and c = 16.

The equation to the circle is then

x'^ + y^ + 2gx-8y + 16 = 0.

This meets the axis of x in points given by

x'^ + 2gx + l&:=0,

i.e. at points distant

-g+slT^^ and -g-Jg^lJG,

Hence Q = 2^g^-ie.

Therefore ^= Â±5, and the required equation is

x^ + y^Â±10x-8y + U = 0.

There are therefore two circles satisfying the given conditions.

This is geometrically obvious.

EXAMPLES. XVII.

Find the equation to the circle

1. Whose radius is 3 and whose centre is ( - 1, 2).

2. Whose radius is 10 and whose centre is ( - 5, - 6).

3. Whose radius is a + b and whose centre is {a, - h).

4. Whose radius is Ja^ - b'^ and whose centre is { - a, - 6).

Find the^ coordinates of the centres and the radii of the circles

whose equations are

5. x^ + y^-ix-8y = ^l. Q, Sx'^ + Sz/ - 5x- Qy + 4 = 0.

124 COORDINATE GEOMETRY. [ExS.

7. x^ + y^=k{x + k). 8. x^ + y'^ = 2gx-2fy.

9. Jl+m"^ {x^ + if) - 2cx - 2mcy = 0.

Draw tlie circles whose equations are

10. x^ + y^ = 2ay. H, 3x^ + 3y^=4x.

12. 5x'^ + 5y^=2x + 3y.

13. Find the equation to the circle which passes through the

points (1, - 2) and (4, - 3) and which has its centre on the straight

line 305 + 42/ = 7.

14. Find the equation to the circle passing through the points

(0, a) and (6, h), and having its centre on the axis of x.

Find the equations to the circles which pass through the points

15. (0, 0), (a, 0), and (0, 6). 16. (1, 2), (3, -4), and (5, -6).

17. (1,1), (2,-1), and (3, 2). 18. (5, 7), (8, 1), and (1, 3).

19. (Â«, &)> (Â«j -&)> and(a + 6, a-b).

20. ABGB is a square whose side is <x; taking AB and AD as

axes, prove that the equation to the circle circumscribing the square is

x^-\-y^ = a{x-\-y).

21. Find the equation to the circle which passes through the

origin and cuts off intercepts equal to 3 and 4 from the axes.

22. Find the equation to the circle passing through the origin

and the points (a, 6) and (6, a). Find the lengths of the chords that

it cuts off from the axes.

23. Find the equation to the circle which goes through the origin

and cuts off intercepts equal to h and h from the positive parts of the

axes.

24. Find the equation to the circle, of radius a, which passes

through the two points on the axis of x which are at a distance & from

the origin.

Find the equation to the circle which

25. touches each axis at a distance 5 from the origin.

26. touches each axis and is of radius a,

27. touches both axes and passes through the point ( - 2, - 3).

28. touches the axis of x and passes through the two points

(1, -2) and (3, -4).

29. touches the axis of y at the origin and passes through the

point (6, c).

XVII.] TANGENT TO A CIRCLE. 125

30. touches the axis of a; at a distance 3 from the origin and

intercepts a distance 6 on the axis of y.

31. Points (1, 0) and (2, 0) are taken on the axis of x, the axes

being rectangular. On the line joining these points an equilateral

triangle is described, its vertex being in the positive quadrant. Find

the equations to the circles described on its sides as diameters.

32. If 1/ = mx be the equation of a chord of a circle whose radius is

a, the origin of coordinates being one extremity of the chord and the

axis of X being a diameter of the circle, prove that the equation of a

circle of which this chord is the diameter is

(1 + m2) (Â£c2 + 1/2) -2a{x + my) = 0.

33. Find the equation to the circle passing through the points

(12, 43), (18, 39), and (42, 3) and prove that it also passes through

the points ( - 54, - 69) and ( - 81, - 38).

34. Find the equation to the circle circumscribing the quadrilateral

formed by the straight lines

2x + 3y = 2, dx-2y = 4:, x + 2y = 3, and 2x-y = B.

35. Prove that the equation to the circle of which the points

{x^ , y^) and {x^ , y^) ^^^ the ends of a chord of a segment containing an

angle 6 is

{x - x^) {x - x^) + (y - yj) (y - y^)

Â± cot 9 [{x - Xj) {y - 2/2) - {x -x^) (y - y^)] = 0.

36. Find the equations to the circles in which the line joining the

points {a, h) and {b, - a) is a chord subtending an angle of 45Â° at any

point on its circumference.

148. Tangent. Euclid in his Book III. defines the

tangent at any point of a circle, and proves that it is always

perpendicular to the radius drawn from the centre to the

point of contact.

From this property may be deduced the equation to the

tangent at any point (x\ y') of the circle x^ ^-y^ â€” o?.

For let the point P (Fig. Art. 139) be the point

(a;', y').

The equation to any straight line passing through T is,

by Art. 62,

y~ y' = m, {x â€” x) (1).

Also the equation to OP is

3/ = |* (2).

126 COORDINATE GEOMETRY.

The straight lines (1) and (2) are at right angles, i.e. the

line (1) is a tangent, if

mx^ = -l, (Art. 69)

I. e. II m = > .

y

Substituting this value of m in (1), the equation of the

tangent at {x\ y) is

y-y = - f{^-^h

t.e. xx+yyâ€” x'^ + y (oj.

But, since {x, y) lies on the circle, we have x^ + y"^ â€” a^,

and the required equation is then

XX' + yy' = a2.

149. In the case of most curves it is impossible to

give a simple construction for the tangent as in the case of

the circle. It is therefore necessary, in general, to give a

different definition.

Tangent. Def. Let F and Q be any two points, near

to one another, on any curve.

Join TQ \ then FQ is called a

secant.

The position of the line PQ when

the point Q is taken indefinitely close

to, and ultimately coincident with, the

point F is called the tangent at F. X ^

The student may better appreciate ^^ - ,___^^

this definition, if he conceive the curve

to be made up of a succession of very small points (much

smaller than could be made by the finest conceivable drawing

pen) packed close to one another along the curve. The

tangent at F is then the straight line joining F and the

next of these small points.

150. To find the equation of the tangent at the imint

{x\ y) of the circle x^ + y^ = a^.

EQUATION TO THE TANGENT. 127

Let P be the given point and Q a point (x\ y") lying on

the curve and close to P.

The equation to FQ is then

y-2/' = f^(Â»-Â»^') (1)-

Since both (cc', y) and {x\ y") lie on the circle, we have

x'^ + y'^ = a\

and x"^ + y'"^ = a^.

By subtraction, we have

x"^-x" + y"'-y" = 0,

i. e. {x" - x') {x" + x') + (y" - y') (y" + y') ^ 0,

// / It , r

y â€”y X + x

X - X y + y

Substituting this value in (1), the equation to FQ is

y-y'=-^^'(''-'') (^)-

Now let Q be taken very close to F, so that it ulti-

mately coincides with P, i, e. put x" = x and y" = y.

Then (2) becomes

2x

y-y' = -7y-A^-^\

^y

i. e. yy' + xx' â€” x'^ + y'^ = a^.

The required equation is therefore

xx' + yy' = a2 (3).

It will be noted that the equation to the tangent

found in this article coincides with the equation found

from Euclid's definition in Art. 148.

Our definition of a tangent and Euclid's definition there-

fore give the same straight line in the case of a circle,

151. To obtain the equation of the tangent at any point

(x'y y') lying on the circle

x^ + y'^ + 2gx + 2fy + c = 0.

128 COORDINATE GEOMETRY.

Let P be the given point and Q a point (x'\ y") lying on

the curve close to P.

The equation to PQ is therefore

2/-2/' = C^!(Â«-x') (1).

Since both (x\ y) and (cc", y") lie on the circle, we have

x'^ + y'^ + 'lgx' + 2fy' + C = (2),

and x"^ + y"' + 2gx" + 2fy" + c^^ (3).

By subtraction, we have

^- _ x'^ + y"^ _ y- + 2g (x" - x') + 2/(y" - y') = 0,

i. e. (x" - x) {x" + x' + 2g) + (y" - y') {y" + y' + 2/) = 0,

y â€”y X + X â€¢{â€¢ Zg

X - X y +y +2/

Substituting this value in (1), the equation to PQ be-

comes

f X + X + Ag . ,. / , V

Now let Q be taken very close to P, so that it ultimately

coincides with P, i. e. put x" â€” x and y" = y'.

The equation (4) then becomes

^. e. y {y +/) + 0^(03' + ^) = y' {y' +f)+x {x' + g)

to fO / /â€¢ /

= a:^ + 2/- + p'aj +fy

= -9^' -fy -(^i

by (2).

This may be written

XX' + yy' + g (x + X') + f (y + y) + c = O

which is the required equation.

152. The equation to the tangent at (x', y') is there-

fore obtained from that of the circle itself by substituting

XX for x^y yy' for y^, x â– {â– x' for 2a;, and y -V y for 2y.

INTERSECTIONS OF A STRAIGHT LINE AND A CIRCLE. 129

This is a particular case of a general rule Avhich will be

found to enable us to write down at sight the equation to

the tangent at ix\ y') to any of the curves with which we

shall deal in this book.

153. Points of intersection, in general, of the straight

line

y^mx -\- G (1),

with the circle x' + y^ = d^ (2).

The coordinates of the points in which the straight line

(1) meets (2) satisfy both equations (1) and (2).

If therefore we solve them as simultaneous equations

we shall obtain the coordinates of the common point or

points.

Substituting for y from (1) in (2), the abscissae of the

required points are given by the equation

x^ + {mx + c)^ = a^,

i.e. x^ (1 + m^) + 2mcx + c^ â€” Â«- = (3).

The roots of this equation are, by Art. 1, real, coinci-

dent, or imaginary, according as

(2mc)^ â€” 4 (1 + m^) (c^ â€” a^) is positive, zero, or negative,

i.e. according as

a? {\ -\- m^) â€” c^ is positive, zero, or negative,

i.e. according as

c^ is < = or > a^ (1 + m^).

In the figure the lines marked I, II, and III are all

parallel, i.e. their equations all have the same "r/i."

L. 9

130 COORDINATE GEOMETRY.

The straight line I corresponds to a value of c^ which

is <a^ (1 + oYv^) and it meets the circle in two real points.

The straight line III which corresponds to a value of c^,

> a^ (1 + m^), does not meet the circle at all, or rather, as in

Art. 108, this is better expressed by saying that it meets

the circle in imaginary points.

The straight line II corresponds to a value of c', which

is equal to a^ (1 + m^), and meets the curve in two coincident

points, i.e. is a tangent.

154. We can now obtain the length of the chord inter-

cepted by the circle on the straight line (1). For, if x^ and

X2 be the roots of the equation (3), we have

2mc _, c^ â€” cC-

^ ^ 1 + m" ^ ^ 1 + m^

Hence

2

^1 â€” ^2 = Jip^i + ^2)^ â€” ^x^x^ = -z "2 Jm^c^ - {(fâ€”a^) (1 + m^)

J. "r 7Yh

1 +m^

If 2/1 and 2/2 ^^ t^6 ordinates of Q and R we have, since

these points are on (1),

2/1 â€” 2/2 = {mx-^ + <^) "~ (^^2 + c) = rri {x^ â€” x^.

Hence

QR = J(yi - y^Y + K - oc^y = Jl+ m' (x^^ - x.^

y

a^ (1 + m^) â€” c^

1 + 7n^

In a similar manner we can consider the points of inter-

section of the straight line y = mx + k with the circle

a;^ + 2/2 + 2yx + 2fy + c = 0.

155. The straight line

y = mx + ajl + m^

is always a. tangent to the circle

2 Q o

X + y â€” a .

EQUATION TO ANY TANGENT. 131

As in Art. 153 the straight line

y = mx + G

meets the circle in two points which are coincident if

But if a straight line meets the circle in two points

which are indefinitely close to one another then, by Art.

149, it is a tangent to the circle.

The straight line y = mx + c is therefore a tangent to the

circle if

G^a J\ + 772^,

i.e. the equation to any tangent to the circle is

y = mx + a Vl + m2 (1).

Since the radical on the right hand may have the + or â€”

sign prefixed we see that corresponding to any value of in

there are two tangents. They are marked II and IV in

the figure of Art. 153.

156. The above result may also be deduced from the equation

(3) of Art. 150, which may be written

x' a2

y= - Â«+- (1).

y y

x'

Put â€” , = 'm, so that x'= -my' , and the relation .r'- + ?/'2 = a^ gives

y'^{m^+l) = a^, i.e. â€”= /Jl+m^.

The equation (1) then becomes

y = mx + afjl + m\

This is therefore the tangent at the point whose coordinates are

~ma . a

and

Vi + w2 7i +

W''

157. If we assume that a tangent to a circle is always perpen-

dicular to the radius vector to the point of contact, the result of

Art. 155 may be obtained in another manner.

For a tangent is a line whose perpendicular distance from the

centre is equal to the radius.

9â€”2

132 COORDINATE GEOMETRY.

The straight line y=mx + c will therefore touch the circle if the

perpendicular on it from the origin be equal to a, i.e. if

i.e. if c = a /sjl + m^.

This method is not however applicable to any other curve besides the

circle.

158. Ex. Find the equations to the tangents to the circle

x'^ + y^-6x + Ay = 12

which are parallel to the straight line

4a;+3i/ + 5 = 0.

Any straight line parallel to the given one is '

4a: + 32/ + (7 = (1).

The equation to the circle is

(a; -3)2 + (2/ + 2)2 =52.

The straight line (1), if it be a tangent, must be therefore such

that its distance from the point (3, - 2) is equal to Â±5.

Hence ^l7^Â±f=:Â±5 (Art. 75),

V4H32

so that (7= -6=t25 = 19or -31.

The required tangents are therefore

4a; + 32/ + 19 = and 4;r + 3?/-31 = 0.

159. Normal. Def. The normal at any point P of

a curve is the straight line which passes through P and is

perpendicular to the tangent at P.

To find the equation to the normal at the j^oint (x', y'^ of

(1) the circle

X- + 2/ = a%

and (2) the circle

X- + y^ + Igx + 2/2/ + c = 0.

(1) The tangent at (.t', 3/') is

XX + yy â€” a^,

X a^

i.e. y^ ,x + â€” .

y y

THE NORMAL TO THE CIRCLE. 133

The equation to the straight line passing through {x\ y)

perpendicular to this tangent is

y-y' ^m{x- x),

where m x (- %] = - 1, (Art. 69),

y'

X

The required equation is therefore

y-y ^-'(^-^-^)' .

i. e. X y â€” xy = 0.

This straight line passes through the centre of the circle

which is the point (0, 0).

If we assume Euclid's propositions the equation is at once

written down, since the normal is the straight line joining

(0, 0) to (Â«;', 2/').

(2) The equation to the tangent at {x ^ y) to the circle

'Â£ + i/-^ + ^yx + %fy + c â€”

X + CI gx + fu + c , ^ T ^ ^ V

IS y = - â€” -.x-^- /-^^ . (Art. 151.)

The equation to the straight line, passing through the

point (a;', ?/') and perpendicular to this tangent, is

y-y =m{x- x'),

where m x f-%^\ - - 1, (Art. 69),

y+f

I.e. m^â€”, â€” - .

x + g

The equation to the normal is therefore

y-y'-^K~^{^-x),

X + g ^ '

i. e. y {x' +g)-x{y +/)+ fx - gy' = 0.

134 COORDINATE GEOMETRY.

EXAMPLES. XVIII.

Write down the equation of the tangent to the circle

1. a;2 + 7/2 -3a; + 10?/ = 15 at the point (4, -11).

2. 4^2 + iy^ - 16a; + 24y -^ 117 at the point ( - 4, - -V) .

Find the equations to the tangents to the circle

3. a;2 + 2/^ = 4 which are parallel to the line a; + 2^ + 3 = 0.

ab-h^ db'-K''

and . o =^ â€” . o / ,

sm" (i) sm^ o)

0) a7id oi' being the angles between the original and final indrs

of axes.

Let the coordinates of any point P, referred to the

original axes, be x and y and, referred to the final axes, let

them be x and y .

By Art. 20 the square of the distance between P and

the origin is o^ + 2xy cos a> + 2/^, referred to the original axes,

and x'^ + 2xy cos w' + y"^^ referred to the final axes.

We therefore always have

X- + Ixy cos w + 2/^ = a;'^ + ^xy cos m -vy''^ (1).

Also, by supposition, we have

ax"" + 2hxy + by"" = ax'- + 2h'xy' + b'lj^ (2).

Multiplying (1) by \ and adding it to (2), we therefore have

x^ (Â« + X) + 2xy (h + X cos to) +y^ (b + X)

= x"" (a' + X) + 2x'y' {h' + X cos o>') + y" {b' + X) . . .(3).

If then any value of X makes the left-hand side of (3) a

perfect square, the same value must make the right-hand

side also a perfect square.

But the values of X which make the left-hand a perfect

square are given by the condition

(h + X cos (o)2 -(a + X) {b + X),

EXAMPLES. 117

i.e. by

\2 (1 - cos^ w) + X{a + h-2h cos oi) + ab- h^ = 0,

^â€ž . a + h â€” 2hcos(ji ab â€” h^^ ,,,

i.e. by X^ + X r^ + -r-^ â€” = (4).

In a similar manner the values of A, which make the

right-hand side of (3) a perfect square are given by the

equation

^a' + b'-2h'cosoy' a'b'-h'^ ^

X' + X r^-^ + . ^ , =0 (5).

sm^ to sm'^ci) ^ '

Since the values of \ given by equation (4) are the same

as the values of X given by (5), the two equations (4) and

(5) must be the same.

Hence we have

a + b â€” 2h cos cu a' + b' â€” 2h' cos co'

and

sin^ (o sin^ co'

ab-h2 a'b'-h'2

sin2 0) sin2 cd'

EXAMPLES. XVI.

1. The equation to a straight line referred to axes inclined at 30Â°

to one another is y = 2x + l. Find its equation referred to axes

inclined at 45Â°, the origin and axis of x being unchanged.

2. Transform the equation 2x'^ + 3 sjSxy + Sy^ = 2 from axes

inclined at 30Â° to rectangular axes, the axis of x remaining

unchanged.

3. Transform the equation x^ + xy + y^ = 8 from axes inclined at

60Â° to axes bisecting the angles between the original axes.

4. Transform the equation y'^-\-4cy cot a - 4cc=0 from rectangular

axes to oblique axes meeting at an angle a, the axis of x being kept

the same.

5. If aj and y be the coordinates of a point referred to a system of

obUque axes, and x' and y' be its coordinates referred to another

system of oblique axes with the same origin, and if the formulae of

transformation be

x=mx' + ny' and y â€” m'x'-\-n'y\

., . m^ + m'^-1 mm'

prove that Â» , ,. ., = â€” 7- .

CHAPTER VIII.

THE CIRCLE.

138. Def. A circle is the locus of a point which

moves so that its distance from a fixed point, called the

centre, is equal to a given distance,

called the radius of the circle.

The given distance is

139. To find the equation to a circle, the axes of coordi-

nates being two straight lines through its centre at right

angles.

Let be the centre of the circle and let a be its radius.

Let OX and OF be the axes of

coordinates.

Let P be any point on the circum-

ference of the circle, and let its coordi-

nates be X and y.

Draw PJf perpendicular to OX and

join OP.

Then (Euc. I. 47)

OM^ + MP' = a',

i.e. x2 + y2 = a2.

This being the relation which holds between the coordi-

nates of any point on the circumference is, by Art. 42, the

required equation.

140. To find the equation to a circle referred to any

rectangular axes.

THE CIRCLE.

119

Let OX and (9 F be the two rectangular axes.

Let C be the centre of the

circle and a its radius.

Take any point P on the

circumference and draw per-

pendiculars CM and FN upon

OX ; let F be the point (x, y).

Draw GL perpendicular to

NF.

Let the coordinates of G be

h and h ; these are supposed to be known.

We have GL = MN= ON- OM=x- h,

and LF = NF-NL = NF-MG^y-h.

Hence, since GL^ + LF^ = GF\

we have (x-h)2+ (y-k)2 = a2 (1).

This is the required equation.

Ex. The equation to the circle, whose centre is the point ( - 3, 4)

and whose radius is 7, is

i.e. a;2 + 2/2 + 6a:-8?/ = 24.

141. Some particular cases of the preceding article may be

noticed :

(a) Let the origin be on the circle so that, in this case,

i.e. h^ + Jc'^=a^.

The equation (1) then becomes

i.e. x'^ + y^-2hx-2ky = 0.

(jS) Let the origin be not on the curve, but let the centre lie on

the axis of x. In this case k = 0, and the equation becomes

{x - lif â– \-y^ = a^.

(7) Let the origin be on the curve and let the axis of re be a

diameter. We now have fe = and a = h, so that the equation becomes

x^ + t/-2hx = 0.

(8) By taking at C, and thus making both h and k Tiero, we

have the case of Art. 139.

120 COORDINATE GEOMETRY.

(e) The circle will touch the axis of x if MG be equal to the

radius, i.e. if k = a.

The equation to a circle touching the axis of x is therefore

x'^ + 7f-2hx-2ky + h'^ = Q.

Similarly, one touching the axis of y is

x^ + y^-2hx-2ky + k^ = 0.

142. To prove that the equation

a^ + 2/'+2^a;+2/2/ + c = (1),

always represents a circle for all values of g,f and c, and to

find its centre and radius. [The axes are assumed to be

rectangular.]

This equation may be written

{a? + 2gx + cf) + {f + 2/2/ +/^) = cf +f' - c,

i.e. {X + gf 4- {y +ff = {JfTp^cf.

Comparing this with the equation (1) of Art. 140, we

see that the equations are the same if

h = -g, h^-f and a = Jg"" -hf^-c.

Hence (1) represents a circle whose centre is the point

(â€” ^, â€”f), and whose radius is Jg^-^f^ â€” c.

If g^ +f' > c, the radius of this circle is real.

If g^ +f'^ = c, the radius vanishes, i. e. the circle becomes

a point coinciding with the point (â€” ^, â€”f). Such a circle

is called a point-circle.

If g^ +f^ < c, the radius of the circle is imaginary. In

this case the equation does not represent any real geo-

metrical locus. It is better not to say that the circle does

not exist, but to say that it is a circle with a real centre

and an imaginary radius.

Ex. 1. The equation x"^ + ij^ + 4^x - 6y = can be written in the

form

{x + 2f + iy-Bf = lS = {JlS)\

and therefore represents a circle whose centre is the point ( - 2, 3) and

whose radius is >,yi3.

GENERAL EQUATION TO A CIRCLE. 121

Ex. 2. The equation 45a;2 + 45?/2 - 60x- + SQy + 19 = is equivalent

to

i.e. (Â«'-|)^ + (y + F=f + ^*5-H=/A,

and therefore represents a circle whose centre is the point (f , -|) and

whose radius is â€”=- .

15

143. Condition that the general equation of the second

degree may represent a circle.

The equation (1) of the preceding article, multiplied by

any arbitrary constant, is a particular case of the general

equation of the second degree (Art. 114) in which there is

no term containing xy and in which the coefficients of x^

and y^ are equal.

The general equation of the second degree in rectangular

coordinates therefore represents a circle if the coefficients

of x^ and y^ be the same and if the coefficient of xy

be zero.

144. The equation (1) of Art. 142 is called the

general equation of a circle^ since it can, by a proper

choice of g, f and c, be made to represent any circle.

The three constants g, f, and c in the general equation

correspond to the geometrical fact that a circle can be found

to satisfy three independent geometrical conditions and no

more. Thus a circle is determined when three points on it

are given, or when it is required to touch three straight

lines.

145. To find the equation to the circle which is described on the

line joining the points {x-^ , y-^) and (x^ , 2/2) ^^ diameter.

Let A be the point (a;^, y-^) and B be the point {x^, 2/2) Â» ^^^ ^^^ ^^^

coordinates of any point P on the circle be h and k.

The equation to AP is (Art. 62)

y-yi=j^i^-^i) (1).

and the equation to BP is

2/ -2/2=^' (^-^2) (2).

But, since APB is a semicircle, the angle APB is a right angle,

and hence the straight Unes (1) and (2) are at right angles.

122

COOKDINATE GEOMETRY.

Hence, by Art. 69, we have

^-Vi _ h^ljh^ _ 1

i.e. (h-x-^){h-x^) + {k-yj}{Jc-y2) = 0.

But this is the condition that the point {h, k) may lie on the curve

whose equation is

{X - x^) {x - x^) + {y- y^) {y - y^) = 0.

This therefore is the required equation.

146. Intercepts made on the axes by the circle whose equation is

ax^ + ay^ + 2gx + 2fy + c = (1).

The abscissae of the points where the circle (1) meets the axis of x,

i.e. y = 0, are given by the equation

ax^ + 2gx + c = (2).

The roots of this equation being Xj^ and x^ ,

we have

^ + â€¢^2

29

a 'â–

and

(Art. 2.

Hence

A-^A^ = x^ -x-^= J{x-^-\-x^^ - 4x-^i

V a2

= 2

a a

Again, the roots of the equation (2) are both imaginary if g^<ac.

In this case the circle does not meet the axis of x in real points, i.e.

geometrically it does not meet the axis of x at all.

The circle will touch the axis of x if the intercept A-^A^, be just

zero, i.e. ii g^ = ac.

It will meet the axis of x in two points lying on opposite sides of

the origin if the two roots of the equation (2) are of opposite signs,

i. e. if c be negative.

147. Ex.1. Find the equation to the circle which passes through

the points (1, 0), (0, - 6), and (3, 4).

Let the equation to the circle be

x^ + y^ + 2gx + 2fy + c = (1).

Since the three points, whose coordinates are given, satisfy this

equation, we have

l + 2^ + c = (2),

36-12/+c=0 (3),

and 25 + 6^ + 8/+c = (4).

EXAMPLES. 123

Subtracting (2) from (3) and (3) from (4), we have

2^ + 12/= 35,

and 65f + 20/=ll.

Hence /=*/- and g=z -iJ-.

Equation (2) then gives c = ^-.

Substituting these values in (1) the required equation is

4a;2 + 4i/2 - 142a; + 47a; + 138 = 0.

Ex. 2. Find the equation to the circle which touches the axis ofy

at a distance + 4 from the origin and cuts off an intercept 6 from the

axis of X.

Any circle is x^ + y'^ + 2gx + 2fy + c = 0.

This meets the axis of y in points given by

y^ + 2fy + c = 0.

The roots of this equation must be equal and each equal to 4, so

that it must be equivalent to {y - 4)^ = 0.

Hence 2/= -8, and c = 16.

The equation to the circle is then

x'^ + y^ + 2gx-8y + 16 = 0.

This meets the axis of x in points given by

x'^ + 2gx + l&:=0,

i.e. at points distant

-g+slT^^ and -g-Jg^lJG,

Hence Q = 2^g^-ie.

Therefore ^= Â±5, and the required equation is

x^ + y^Â±10x-8y + U = 0.

There are therefore two circles satisfying the given conditions.

This is geometrically obvious.

EXAMPLES. XVII.

Find the equation to the circle

1. Whose radius is 3 and whose centre is ( - 1, 2).

2. Whose radius is 10 and whose centre is ( - 5, - 6).

3. Whose radius is a + b and whose centre is {a, - h).

4. Whose radius is Ja^ - b'^ and whose centre is { - a, - 6).

Find the^ coordinates of the centres and the radii of the circles

whose equations are

5. x^ + y^-ix-8y = ^l. Q, Sx'^ + Sz/ - 5x- Qy + 4 = 0.

124 COORDINATE GEOMETRY. [ExS.

7. x^ + y^=k{x + k). 8. x^ + y'^ = 2gx-2fy.

9. Jl+m"^ {x^ + if) - 2cx - 2mcy = 0.

Draw tlie circles whose equations are

10. x^ + y^ = 2ay. H, 3x^ + 3y^=4x.

12. 5x'^ + 5y^=2x + 3y.

13. Find the equation to the circle which passes through the

points (1, - 2) and (4, - 3) and which has its centre on the straight

line 305 + 42/ = 7.

14. Find the equation to the circle passing through the points

(0, a) and (6, h), and having its centre on the axis of x.

Find the equations to the circles which pass through the points

15. (0, 0), (a, 0), and (0, 6). 16. (1, 2), (3, -4), and (5, -6).

17. (1,1), (2,-1), and (3, 2). 18. (5, 7), (8, 1), and (1, 3).

19. (Â«, &)> (Â«j -&)> and(a + 6, a-b).

20. ABGB is a square whose side is <x; taking AB and AD as

axes, prove that the equation to the circle circumscribing the square is

x^-\-y^ = a{x-\-y).

21. Find the equation to the circle which passes through the

origin and cuts off intercepts equal to 3 and 4 from the axes.

22. Find the equation to the circle passing through the origin

and the points (a, 6) and (6, a). Find the lengths of the chords that

it cuts off from the axes.

23. Find the equation to the circle which goes through the origin

and cuts off intercepts equal to h and h from the positive parts of the

axes.

24. Find the equation to the circle, of radius a, which passes

through the two points on the axis of x which are at a distance & from

the origin.

Find the equation to the circle which

25. touches each axis at a distance 5 from the origin.

26. touches each axis and is of radius a,

27. touches both axes and passes through the point ( - 2, - 3).

28. touches the axis of x and passes through the two points

(1, -2) and (3, -4).

29. touches the axis of y at the origin and passes through the

point (6, c).

XVII.] TANGENT TO A CIRCLE. 125

30. touches the axis of a; at a distance 3 from the origin and

intercepts a distance 6 on the axis of y.

31. Points (1, 0) and (2, 0) are taken on the axis of x, the axes

being rectangular. On the line joining these points an equilateral

triangle is described, its vertex being in the positive quadrant. Find

the equations to the circles described on its sides as diameters.

32. If 1/ = mx be the equation of a chord of a circle whose radius is

a, the origin of coordinates being one extremity of the chord and the

axis of X being a diameter of the circle, prove that the equation of a

circle of which this chord is the diameter is

(1 + m2) (Â£c2 + 1/2) -2a{x + my) = 0.

33. Find the equation to the circle passing through the points

(12, 43), (18, 39), and (42, 3) and prove that it also passes through

the points ( - 54, - 69) and ( - 81, - 38).

34. Find the equation to the circle circumscribing the quadrilateral

formed by the straight lines

2x + 3y = 2, dx-2y = 4:, x + 2y = 3, and 2x-y = B.

35. Prove that the equation to the circle of which the points

{x^ , y^) and {x^ , y^) ^^^ the ends of a chord of a segment containing an

angle 6 is

{x - x^) {x - x^) + (y - yj) (y - y^)

Â± cot 9 [{x - Xj) {y - 2/2) - {x -x^) (y - y^)] = 0.

36. Find the equations to the circles in which the line joining the

points {a, h) and {b, - a) is a chord subtending an angle of 45Â° at any

point on its circumference.

148. Tangent. Euclid in his Book III. defines the

tangent at any point of a circle, and proves that it is always

perpendicular to the radius drawn from the centre to the

point of contact.

From this property may be deduced the equation to the

tangent at any point (x\ y') of the circle x^ ^-y^ â€” o?.

For let the point P (Fig. Art. 139) be the point

(a;', y').

The equation to any straight line passing through T is,

by Art. 62,

y~ y' = m, {x â€” x) (1).

Also the equation to OP is

3/ = |* (2).

126 COORDINATE GEOMETRY.

The straight lines (1) and (2) are at right angles, i.e. the

line (1) is a tangent, if

mx^ = -l, (Art. 69)

I. e. II m = > .

y

Substituting this value of m in (1), the equation of the

tangent at {x\ y) is

y-y = - f{^-^h

t.e. xx+yyâ€” x'^ + y (oj.

But, since {x, y) lies on the circle, we have x^ + y"^ â€” a^,

and the required equation is then

XX' + yy' = a2.

149. In the case of most curves it is impossible to

give a simple construction for the tangent as in the case of

the circle. It is therefore necessary, in general, to give a

different definition.

Tangent. Def. Let F and Q be any two points, near

to one another, on any curve.

Join TQ \ then FQ is called a

secant.

The position of the line PQ when

the point Q is taken indefinitely close

to, and ultimately coincident with, the

point F is called the tangent at F. X ^

The student may better appreciate ^^ - ,___^^

this definition, if he conceive the curve

to be made up of a succession of very small points (much

smaller than could be made by the finest conceivable drawing

pen) packed close to one another along the curve. The

tangent at F is then the straight line joining F and the

next of these small points.

150. To find the equation of the tangent at the imint

{x\ y) of the circle x^ + y^ = a^.

EQUATION TO THE TANGENT. 127

Let P be the given point and Q a point (x\ y") lying on

the curve and close to P.

The equation to FQ is then

y-2/' = f^(Â»-Â»^') (1)-

Since both (cc', y) and {x\ y") lie on the circle, we have

x'^ + y'^ = a\

and x"^ + y'"^ = a^.

By subtraction, we have

x"^-x" + y"'-y" = 0,

i. e. {x" - x') {x" + x') + (y" - y') (y" + y') ^ 0,

// / It , r

y â€”y X + x

X - X y + y

Substituting this value in (1), the equation to FQ is

y-y'=-^^'(''-'') (^)-

Now let Q be taken very close to F, so that it ulti-

mately coincides with P, i, e. put x" = x and y" = y.

Then (2) becomes

2x

y-y' = -7y-A^-^\

^y

i. e. yy' + xx' â€” x'^ + y'^ = a^.

The required equation is therefore

xx' + yy' = a2 (3).

It will be noted that the equation to the tangent

found in this article coincides with the equation found

from Euclid's definition in Art. 148.

Our definition of a tangent and Euclid's definition there-

fore give the same straight line in the case of a circle,

151. To obtain the equation of the tangent at any point

(x'y y') lying on the circle

x^ + y'^ + 2gx + 2fy + c = 0.

128 COORDINATE GEOMETRY.

Let P be the given point and Q a point (x'\ y") lying on

the curve close to P.

The equation to PQ is therefore

2/-2/' = C^!(Â«-x') (1).

Since both (x\ y) and (cc", y") lie on the circle, we have

x'^ + y'^ + 'lgx' + 2fy' + C = (2),

and x"^ + y"' + 2gx" + 2fy" + c^^ (3).

By subtraction, we have

^- _ x'^ + y"^ _ y- + 2g (x" - x') + 2/(y" - y') = 0,

i. e. (x" - x) {x" + x' + 2g) + (y" - y') {y" + y' + 2/) = 0,

y â€”y X + X â€¢{â€¢ Zg

X - X y +y +2/

Substituting this value in (1), the equation to PQ be-

comes

f X + X + Ag . ,. / , V

Now let Q be taken very close to P, so that it ultimately

coincides with P, i. e. put x" â€” x and y" = y'.

The equation (4) then becomes

^. e. y {y +/) + 0^(03' + ^) = y' {y' +f)+x {x' + g)

to fO / /â€¢ /

= a:^ + 2/- + p'aj +fy

= -9^' -fy -(^i

by (2).

This may be written

XX' + yy' + g (x + X') + f (y + y) + c = O

which is the required equation.

152. The equation to the tangent at (x', y') is there-

fore obtained from that of the circle itself by substituting

XX for x^y yy' for y^, x â– {â– x' for 2a;, and y -V y for 2y.

INTERSECTIONS OF A STRAIGHT LINE AND A CIRCLE. 129

This is a particular case of a general rule Avhich will be

found to enable us to write down at sight the equation to

the tangent at ix\ y') to any of the curves with which we

shall deal in this book.

153. Points of intersection, in general, of the straight

line

y^mx -\- G (1),

with the circle x' + y^ = d^ (2).

The coordinates of the points in which the straight line

(1) meets (2) satisfy both equations (1) and (2).

If therefore we solve them as simultaneous equations

we shall obtain the coordinates of the common point or

points.

Substituting for y from (1) in (2), the abscissae of the

required points are given by the equation

x^ + {mx + c)^ = a^,

i.e. x^ (1 + m^) + 2mcx + c^ â€” Â«- = (3).

The roots of this equation are, by Art. 1, real, coinci-

dent, or imaginary, according as

(2mc)^ â€” 4 (1 + m^) (c^ â€” a^) is positive, zero, or negative,

i.e. according as

a? {\ -\- m^) â€” c^ is positive, zero, or negative,

i.e. according as

c^ is < = or > a^ (1 + m^).

In the figure the lines marked I, II, and III are all

parallel, i.e. their equations all have the same "r/i."

L. 9

130 COORDINATE GEOMETRY.

The straight line I corresponds to a value of c^ which

is <a^ (1 + oYv^) and it meets the circle in two real points.

The straight line III which corresponds to a value of c^,

> a^ (1 + m^), does not meet the circle at all, or rather, as in

Art. 108, this is better expressed by saying that it meets

the circle in imaginary points.

The straight line II corresponds to a value of c', which

is equal to a^ (1 + m^), and meets the curve in two coincident

points, i.e. is a tangent.

154. We can now obtain the length of the chord inter-

cepted by the circle on the straight line (1). For, if x^ and

X2 be the roots of the equation (3), we have

2mc _, c^ â€” cC-

^ ^ 1 + m" ^ ^ 1 + m^

Hence

2

^1 â€” ^2 = Jip^i + ^2)^ â€” ^x^x^ = -z "2 Jm^c^ - {(fâ€”a^) (1 + m^)

J. "r 7Yh

1 +m^

If 2/1 and 2/2 ^^ t^6 ordinates of Q and R we have, since

these points are on (1),

2/1 â€” 2/2 = {mx-^ + <^) "~ (^^2 + c) = rri {x^ â€” x^.

Hence

QR = J(yi - y^Y + K - oc^y = Jl+ m' (x^^ - x.^

y

a^ (1 + m^) â€” c^

1 + 7n^

In a similar manner we can consider the points of inter-

section of the straight line y = mx + k with the circle

a;^ + 2/2 + 2yx + 2fy + c = 0.

155. The straight line

y = mx + ajl + m^

is always a. tangent to the circle

2 Q o

X + y â€” a .

EQUATION TO ANY TANGENT. 131

As in Art. 153 the straight line

y = mx + G

meets the circle in two points which are coincident if

But if a straight line meets the circle in two points

which are indefinitely close to one another then, by Art.

149, it is a tangent to the circle.

The straight line y = mx + c is therefore a tangent to the

circle if

G^a J\ + 772^,

i.e. the equation to any tangent to the circle is

y = mx + a Vl + m2 (1).

Since the radical on the right hand may have the + or â€”

sign prefixed we see that corresponding to any value of in

there are two tangents. They are marked II and IV in

the figure of Art. 153.

156. The above result may also be deduced from the equation

(3) of Art. 150, which may be written

x' a2

y= - Â«+- (1).

y y

x'

Put â€” , = 'm, so that x'= -my' , and the relation .r'- + ?/'2 = a^ gives

y'^{m^+l) = a^, i.e. â€”= /Jl+m^.

The equation (1) then becomes

y = mx + afjl + m\

This is therefore the tangent at the point whose coordinates are

~ma . a

and

Vi + w2 7i +

W''

157. If we assume that a tangent to a circle is always perpen-

dicular to the radius vector to the point of contact, the result of

Art. 155 may be obtained in another manner.

For a tangent is a line whose perpendicular distance from the

centre is equal to the radius.

9â€”2

132 COORDINATE GEOMETRY.

The straight line y=mx + c will therefore touch the circle if the

perpendicular on it from the origin be equal to a, i.e. if

i.e. if c = a /sjl + m^.

This method is not however applicable to any other curve besides the

circle.

158. Ex. Find the equations to the tangents to the circle

x'^ + y^-6x + Ay = 12

which are parallel to the straight line

4a;+3i/ + 5 = 0.

Any straight line parallel to the given one is '

4a: + 32/ + (7 = (1).

The equation to the circle is

(a; -3)2 + (2/ + 2)2 =52.

The straight line (1), if it be a tangent, must be therefore such

that its distance from the point (3, - 2) is equal to Â±5.

Hence ^l7^Â±f=:Â±5 (Art. 75),

V4H32

so that (7= -6=t25 = 19or -31.

The required tangents are therefore

4a; + 32/ + 19 = and 4;r + 3?/-31 = 0.

159. Normal. Def. The normal at any point P of

a curve is the straight line which passes through P and is

perpendicular to the tangent at P.

To find the equation to the normal at the j^oint (x', y'^ of

(1) the circle

X- + 2/ = a%

and (2) the circle

X- + y^ + Igx + 2/2/ + c = 0.

(1) The tangent at (.t', 3/') is

XX + yy â€” a^,

X a^

i.e. y^ ,x + â€” .

y y

THE NORMAL TO THE CIRCLE. 133

The equation to the straight line passing through {x\ y)

perpendicular to this tangent is

y-y' ^m{x- x),

where m x (- %] = - 1, (Art. 69),

y'

X

The required equation is therefore

y-y ^-'(^-^-^)' .

i. e. X y â€” xy = 0.

This straight line passes through the centre of the circle

which is the point (0, 0).

If we assume Euclid's propositions the equation is at once

written down, since the normal is the straight line joining

(0, 0) to (Â«;', 2/').

(2) The equation to the tangent at {x ^ y) to the circle

'Â£ + i/-^ + ^yx + %fy + c â€”

X + CI gx + fu + c , ^ T ^ ^ V

IS y = - â€” -.x-^- /-^^ . (Art. 151.)

The equation to the straight line, passing through the

point (a;', ?/') and perpendicular to this tangent, is

y-y =m{x- x'),

where m x f-%^\ - - 1, (Art. 69),

y+f

I.e. m^â€”, â€” - .

x + g

The equation to the normal is therefore

y-y'-^K~^{^-x),

X + g ^ '

i. e. y {x' +g)-x{y +/)+ fx - gy' = 0.

134 COORDINATE GEOMETRY.

EXAMPLES. XVIII.

Write down the equation of the tangent to the circle

1. a;2 + 7/2 -3a; + 10?/ = 15 at the point (4, -11).

2. 4^2 + iy^ - 16a; + 24y -^ 117 at the point ( - 4, - -V) .

Find the equations to the tangents to the circle

3. a;2 + 2/^ = 4 which are parallel to the line a; + 2^ + 3 = 0.

Online Library → S. L. (Sidney Luxton) Loney → The elements of coordinate geometry → online text (page 8 of 26)