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the three points (4, 3), (-4, 3), and (0, -5); prove also that the

points so determined lie on a straight line.

12. Find the coordinates of the point of intersection of the

straight lines

2x-3y=^l and 5y-x = S,

and determine also the angle at which they cut one another.

13. Find the angle between the two lines

Sx + y + 12 = and x + 2y-l = 0.

Find also the coordinates of their point of intersection and the

equations of lines drawn perpendicular to them from the point

(3, -2).

VIII.] EXAMPLES. 63

14. Prove that the points whose coordinates are respectively

(5, 1), (1, -1), and (11, 4) lie on a straight line, and find its intercepts

on the axes.

Prove that the following sets of three lines meet in a point.

15. 2x-Sy = 7, Sx-4:y = 13, and 8x-lly = S3.

16. dx + 4.y + G = 0, 6x + 5y + 9 = 0, and Sx + Sy + 5 = 0.

17. - + 7 = 1, j+^ = l, and y = x.

abba

18. Prove that the three straight lines whose equations are

15a;- 18?/ + 1 = 0, 12x + lOi/ - 3 = 0, and 6x + QQy-ll =

all meet in a point.

Shew also that the third line bisects the angle between the other

two.

19. Find the conditions that the straight lines

y = m-^x + ai, y = m^-\-a^, and y = m2X-\-a^

may meet in a point.

Find the coordinates of the orthocentre of the triangles whose

angular points are

20. (0,0), (2, -1), and (-1,3).

21. (1,0), (2,-4), and (-5,-2).

22. In any triangle ABG^ prove that

(1) the bisectors of the angles A, B, and C meet in a point,

(2) the medians, i.e. the lines joining each vertex to the middle

point of the opposite side, meet in a point,

and (3) the straight lines through the middle points of the sides

perpendicular to the sides meet in a point.

Find the equation to the straight line passing through

23. tlie point (3, 2) and the point of intersection of the lines

2x + Sy = l and Sx-Ay = Q.

24. the point (2, - 9) and the intersection of the lines

2x + 5y-8 = and 3x-4y=^S5.

25. the origin and the point of intersection of

x~y-4i=0 and lx + y + 20=0,

proving that it bisects the angle between them.

26. the origin and the point of intersection of the lines

X y ^ ^ X y ^

- + f = 1 and Y + ^ = 1.

a b b a

27. the point (a, b) and the intersection of the same two lines.

28. the intersection of the lines

x-2y-a=0 and x + 3y-2a =

64 COORDINATE GEOMETRY. [Exs.

and parallel to the straight line

29. the intersection of the lines

x + 2y + S = and 3x + iy + 7 =

and perpendicular to the straight line

y-x = 8.

30. the intersection of the lines

dx-iy + l = and 5x + y -1=0

and cutting off equal intercepts from the axes.

31. the intersection of the lines

2x~By = 10 and x + 2y:=Q

and the intersection of the lines

16a;-102/ = 33 and 12x + Uy + 29 = 0.

32. If through the angular points of a triangle straight lines be

drawn parallel to the sides, and if the intersections of these Hnes be

joined to the opposite angular points of the triangle, shew that the

joining lines so obtained will meet in a point.

33. Find the equations to the straight lines passing through the

point of intersection of the straight lines

Ax + By + C = and A'x + B'y + C'^0 and

(1) passing through the origin,

(2) parallel to the axis of y,

(3) cutting off a given distance a from the axis of y,

and (4) passing through a given point {x', y').

34. Prove that the diagonals of the parallelogram formed by the

four straight lines

^?>x + y = 0, ^?>y + x=^0, jBx + y = l, and JBy + x = l

are at right angles to one another.

35. Prove the same property for the parallelogram whose sides

are

- + 7=1, r + - = l, - + | = 2, and t + - = 2.

a b a a o a

36. One side of a square is inclined to the axis of x at an angle a

and one of its extremities is at the origin ; prove that the equations

to its diagonals are

y (cos a - sin a) = x (sin a + cos a)

and ?/ (sin a + cos a) + a: (cos a -sin a) = a.

Find the equations to the straight lines bisecting the angles

between the following pairs of straight lines, placing first the bisector

of the angle in which the origin lies.

37. x+ysJB = %-\-2JB and a;-?/ ^3 = 6-2^3.

VIII.] EXAMPLES. 65

38. 12x + 5y-4c = and Bx+4.tj + 7 = 0.

39. ix + Sij -7 = and 24^ + 7ij- 31 = 0.

40. 2x + y=4: and y + Sx = 5.

41. y-b=^ i>(^-Â«) and y-h = z ^(rc-a).

^ l-m2^ ' ^ 1-m'^^ '

Find the equations to the bisectors of the internal angles of the

triangles the equations of whose sides are respectively

42. 3x + 4:y=e, 12x-5y=B, and 4:X-3y + 12 = 0.

43. Sx + 5y=15, x + y=4:, and 2x + y = Q.

44. Find the equations to the straight lines passing through the

foot of the perpendLcular from the point {h, Jc) upon the straight line

Ax + By + G = and bisecting the angles between the perpendicular

and the given straight line.

45. Find the direction in which a straight Kne must be drawn

through the point (1, 2), so that its point of intersection with the line

x + y = 4: may be at a distance ^^6 from this point.

CHAPTER V.

THE STRAIGHT LINE {continued).

POLAR EQUATIONS. OBLIQUE COORDINATES.

MISCELLANEOUS PROBLEMS. LOCI.

88. To find the general equation to a straight line in

polar coordinates.

Let p be the length of the perpendicular Y from the

origin upon the straight line, and

let this perpendicular make an

angle a with the initial line.

Let P be any point on the

line and let its coordinates be r

and 6.

The equation required will

then be the relation between r, 6, p, and a.

From the triangle YP we have

p = r cos YOP = rcos{a-6)=^r cos (6 - a).

The required equation is therefore

r cos (6 â€” a) =p.

[On transforming to Cartesian coordinates this equation becomes

the equation of Art. 53.]

89. To find the polar equation of the straight line

joining the poiiits whose coordinates are (r^, 6^) and {r^, 6^.

THE STRAIGHT LINE. OBLIQUE COORDINATES. 67

Let A and B be the two given points and P any point

on the line joining them

whose coordinates are r and

e.

Then, since

AA0B-=AA0P+A POB,

we have

^.e.

I.e.

J r^r^ sin A OB = J r^r sin AOP + ^ ri\ sin POB,

r^r^ sin {6 2 â€” 6^ = r^r sin {0 â€” 6-^ + rr^ sin (0^ â€” 6),

sin (^,-^i) _ sin ((9-^1) sin ((92-^)

OBLIQUE COORDINATES.

90. In the previous chapter we took the axes to be

rectangular. In the great majority of cases rectangular

axes are employed, but in some cases oblique axes may be

used with advantage.

In the following articles we shall consider the proposi-

tions in which the results for oblique axes are different

from those for rectangular axes. The propositions of Arts.

50 and 62 are true for oblique, as well as rectangular,

coordinates.

91. To find the equation to a straight line referred to

axes inclined at an angle w.

Let LPL' be a straight line which cuts the axis of

a distance c from the origin and is

inclined at an angle to the axis

of X.

Let P be any point on the

straight line. Draw PNM parallel

to the axis of y to meet OX in M^

and let it meet the . straight line

through C parallel to the axis of x

in the point N,

Let P be the point (.r, y\ so that

CN^OM^x, and NP = MP- 00 ^y-c.

5-

Fat

68 COORDINATE GEOMETRY.

Since L GPN= l FNN' - l PCN' = w - ^, we have

y-c NP _ B>inNC P _ sinO

~ir ~ 'CN~ ^^PN~ sin (o)- ^) â€¢

TT Si^^ /1\

Hence y = x-. â€” ; 7:. + c (1).

^ sm(o>-^) ^ ^

3?his equation is of the form

y = mx + c,

where

sin^ sin^ tan^

7)1 =

sin (o) â€” 6) sin w cos â€” cos to sin sin <o â€” cos w tan ^ '

, , â€ž J /> ^ sin CD

and thererore tan v = .

1 + 7)1 cos a>

In oblique coordinates the equation

y = mx + c

therefore represents a straight line which is inclined at an

angle

- m sin o)

tan~i

l+mcosco

to the axis of x.

Cor. From. (1), by putting in succession equal to 90Â°

and 90Â° + o), we see that the equations to the straight lines,

passing through the origin and perpendicular to the axes of

X and y, are respectively y = and y = â€”x cos w.

92. The axes being oblique, to find the equation to the

straight line, such that the 'perpendicular on it from the origin

is of length p and makes angles a and ^ with the axes of x

and y.

Let LM be the given straight line and OK the perpen-

dicular on it from the origin.

Let P be any point on the

straight line ; draw the ordinate

PN and draw NP perpendicular

to OK and PS "oerpendicular to

NR.

Let P be the point {x, y), so

that OJV = X and NP = y.

THE STRAIGHT LINE. OBLIQUE COORDINATES. 69

The lines NP and Y are parallel.

Also OK and SP are parallel, each being perpendicular

to NB.

Thus lSPN^lKOM=^.

We therefore have

'p = OK- OR + SP = OiVcos a + NP cos p=^xcosa + y cos ^.

Hence x cos a + y cos /5 â€” p = 0,

being the relation which holds between the coordinates of

any point on the straight line, is the required equation.

93. To find the angle between the straight lines

y = mx + c and y = mx + c,

the axes being oblique.

If these straight lines be respectively inclined at angles

and 0' to the axis of x, we have, by the last article,

^ msinoD , ^ ,,, m'sinw

tan u = :. and tan u â€”

1 + ni cos CO 1 + 711 cos 0)

The angle required is 0~ 0'.

XT J. ir\ t\t\ tan ^- tan ^'

Now tan(e-e)=j^:^^^^^-^^,

m sm CD in sm co

1 +m cos CO 1 + 771 cos CO

m sin CO m' sin co

1 +

1+771 cos CO 1 + m' cos CO

_ m sin CO ( 1 + m' cos co) â€” m' sin co (1 + 7?i. cos co)

(1 + m cos co) (1 + W cos co) + ?92m' sin^ co

_ (m â€” w') sin CO

1 + (m + m ) cos CO + mm' '

The required angle is therefore

tan

_j (m â€” in!) sin co

1 + {m + TTb) cos 0) + 7f}im' '

Cor. 1. The two given lines are parallel if m = m'.

Cor. 2. The two given lines are perpendicular if

1 + (m + m') cos 0} + mm' = O.

70 COORDINATE GEOMETRY.

94. If the straight lines have their equations in the

form

Ax-\-By + G = and A'x + B'y â– \- C = 0,

then 7n = - ^ and m = - Wt'

Substituting these values in the result of the last article

the angle between the two lines is easily found to be

J A'B-AB' .

AA' + BB' - {AB' + A'B) cos w

The given lines are therefore parallel if

A'B-AB'=^0.

They are perpendicular if

AA' + BB' = {AB' + A'B) cos w.

95. Ex. The axes being inclined at an angle of 30Â°, obtain the

equations to the straight lines ivhich pass through the origin and are

inclined at 45Â° to the straight line x + y = l.

Let either of the required straight lines be y = mx.

The given straight line isy= -x + 1, so that m'= - 1.

We therefore have

1 + (m + m') cos <a + mm'

where m'= - 1 and w = 30Â°.

rw,- . â– â– â– m+1 . -

This equation gives 2 + (,,,_ 1)^3- 2m = "^^^

Taking the upper sign we obtain m= â€” j^.

Taking the lower sign we have m = - ^3.

The required equations are therefore

y=-sjSx and y=z - ^x,

i.e. y + iJ3x=^0 and JSy + x = 0.

96. To find the length of the lyerpendicular froin the

point (x'j y) upon the straight line Ax + By +0 â€” 0, the axes

being inclined at an angle <a, and the equation being written

so that C is a negative quantity.

THE STRAIGHT LINE. OBLIQUE COORDINATES. 71

Let the given straight line meet the axes in L and M^

G C

so that OL = - , and OM^ â€” -=, .

A B

Let P be the given point {x', y').

Draw the perpendiculars PQ, PR,

and PS on the given line and the

two axes.

Taking and P on opposite sides

of the given line, we then have

IiLPM + AMOL=aOLP + aOPM,

i.e. PQ . LM + OL . OM silicon OL . PE + OM . PS. ..{I).

Draw PU and PV parallel to the axes of y and x, so

that PU = y' and PV-^x.

Hence PE ^ PU sin PUR = y' sin w,

and PS =PV sin P VS - x sin w.

Also

LM= s/OL^ + OM^ - 20 L . Oif cos w

V

(72 C'

0^

z^-'^-^zg'"'"^

- (7 /l+l

2 cos w

since C is a negative quantity.

On substituting these values in (1), we have

pc><(-o)xy-i+-i

2 cos CO C^ .

c

c

so that

PQ =

- 7.7/ sm o> â€” Pi . Â£c sm 00,

Ax^ + By' + C

VA2 + B2 - 2 AB cos CO

. sm a;.

Cor. If (0 â€” 90Â°, i.e. if the axes be rectangular, we

have the result of Art. 75.

72 COORDINATE GEOMETRY.

EXAMPLES. IX.

1. The axes being inclined at an angle of 60Â°, find the inclination

to the axis of x of the straight lines whose equations are

(1) 2/=2^ + 5,

and (2) 2y=.{^^-l)x + l.

2. The axes being inclined at an angle of 120Â°, find the tangent

of the angle between the two straight lines

Qx + ly=.l and 28a; - 73?/ = 101.

3. With oblique coordinates find the tangent of the angle

between the straight lines

y = mx + c and my+x = d.

4. liy=x tan -â€” - and y=x tan â€” j represent two straight lines

at right angles, prove that the angle between the axes is ~ .

5. Prove that the straight lines y + x=c and y=x + d are at

right angles, whatever be the angle between the axes.

6. Prove that the equation to the straight line which passes

through the point {h, Jc) and is perpendicular to the axis of x is

x + y cos o) = h+k cos w.

7. Find the equations to the sides and diagonals of a regular

hexagon, two of its sides, which meet in a corner, being the axes of

coordinates.

8. From each corner of a parallelogram a perpendicular is drawn

upon the diagonal which does not pass through that corner and these

are produced to form another parallelogram ; shew that its diagonals

are perpendicular to the sides of the first parallelogram and that they

both have the same centre.

9. If the straight lines y = miX + Cj^ and y=m,^x + C2 make equal

angles with the axis of x and be not pariallel to one another, prove

that iiij^ + ^2 + 2mjin2 cos w = 0.

10. The axes being inclined at an angle of 30Â°, find the equation

to the straight line which passes through the point ( - 2, 3) and is

perpendicular to the straight line y + Bx = 6.

11. Find the length of the perpendicular drawn from the point

(4, -3) upon the straight line 6a; + 3^ -10 = 0, the angle between the

axes being 60Â°.

12. Find the equation to, and the length of, the perpendicular

drawn from the point (1, 1) upon the straight line 3a; + 4?/ + 5 = 0, the

angle between the axes being 120Â°.

[EXS. IX.] THE STRAIGHT LINE. PROBLEMS. 73

13. The coordinates of a point P referred to axes meeting at an

angle w are [h, k) ; prove that the length of the straight line joining

the feet of the perpendiculars from P upon the axes is

sin w ^y/i^ +k- + 2hk cos w.

14. From a given point {h, k) perpendiculars are drawn to the

axes, whose inclination is oj, and their feet are joined. Prove that

the length of the perpendicular drawn from [h, Jc) upon this line is

hk sin^ 0}

JhF+W+2hkcos^'

and that its equation is hx - ky = h^- k^.

Straight lines passing through fixed points.

97. If the equation to a straic/ht line be of the form

ax + hy + c + \ (ax + b'y + g') = (1 ),

where \ is any arbitrary constant^ it always passes through

one fixed point whatever be the value of \.

For the equation (1) is satisfied by the coordinates of

the point which satisfies both of the equations

ax + by + G~Oj

and a'x 4- b'y + c' = 0.

This point is, by Art. 77,

'be' â€” b'c ca' â€” c'a^

^ab' â€” a'b ' ab' â€” a'b/

and these coordinates are independent of A.

Ex. Given the vertical angle of a triangle in magnitude and

position, and also the sum of the reciprocals of the sides ivhich contain

it; shew that the base always passes through a fixed point.

Take the fixed angular point as origin and the directions of the

sides containing it as axes ; let the lengths of these sides in any such

triangle be a and &, which are not therefore given.

We have - + 7=const. = r- (say) (1).

do K

The equation to the base is

^ y -r

â€” v- -=1

a

z...,by(l), ^'+^(1_1)^1,

1 y

i.e. -(x-y) + ^-l = 0.

74

COORDINATE GEOMETRY.

"Whatever be the value of a this straight line always passes through

the point given by

x-y = and |-1 = 0,

i.e. through ih.e fixed point {k, k).

k

98. Prove that the coordinates of the centre of the

circle inscribed in the triangle, whose vertices are the points

(^ij 2/i)> (^2 J 2/2)5 Â«^^G? (a^3, 2/3), are

axi + hx^ + cx^ ay^ + hy.2 + cy^

a+b+G a+b+c '

where a, 6, Â«7ic? c are ^Ae lengths of the sides of the triangle.

Find also the coordinates of the centres of the escribed

circles.

Let ABC be the triangle and let AD and CE be the

bisectors of the angles A and G

and let them meet in 0'.

Then 0' is the required point.

Since AD bisects the angle

BAC we have, by Euc. YI. 3,

^ _DG_ BD + DG _ _a_

BA ~ AG~ BA+'AC~bTc'

so that

ba

(xj.yj)

'(-,,y,)

DG

b + c

Also, since GO' bisects the angle AGD, we have

^' _ JLC _ _5_ _ 6 + c

0'D~ CD~ ba

a

b + <

The point D therefore divides BG in the ratio

BA : AG, i.e. c : b.

Also 0' divides AD in the ratio b + c : a.

Hence, by Art. 22, the coordinates of D are

cxs + bx^ fi ^3 + ^.'/2

c + b

c + b

THE STRAIGHT LINE. PROBLEMS.

Also, by the same article, the coordinates of 0' are

cxo + hxo ,, X cVo + hy.^

(b + c) X â€”^ â€” â€”^ + ax, (b + c)x -^ â€” ^^ + ay.

^ ' G+h T ' G+b ^

and

{b + c) + a (b + c) +

a

ax^ + bx^, + Gx.^ - ay^ 4- by^ + cy^

a + b + G a + b + c

Again, if 0^ be the centre of the escribed circle opposite

to the angle -4, the line COi bisects the exterior angle of

ACB.

Hence (Euc. VI. A) we have

AO, _AC__ b^G

Therefore Oi is the point which divides AD externally in

the ratio b + g : a.

Its coordinates (Art. 22) are therefore

,y . GXo + bx^ ,-, , cy., + byÂ»

(b + c) ^ ^ ^ - ax^ (b + c) -^ â€” ^ - ay,

^ ^ G + b ^ , ^ ' c + b ^^

ana

{b + G) â€” a (^ + c)

a

â€” axi + bx^ + Gx^ - â€” ayi + by. 2 4- Gy^

â€”a+b+G â€”a+b+c

Similarly, it may be shewn that the coordinates of the

escribed circles opposite to B and C are respectively

^aXj^ â€” bx2 + GXs ay^ â€” by^ + cyA

c-

aâ€”b+G ' aâ€”b+c

and /axj^x2-_cxs ay^ + by.^ - Gy ^

\ a + b â€” c ' a + b â€” G

99. As a numerical example consider the case of the

triangle formed by the straight lines

3x+4cy-7 = 0, 12x + 6y-l7 =0 and 5x + 12y - 34: = 0.

These three straight lines being BC, CA, and AB

respectively we easily obtain, by solving, that the points

A, B, and C are

Q' t)' (t?' 11) ^"'^ (i'^>-

76 COORDINATE GEOMETRY,

Hence

yfi?-')"*(s-')'V

682 5p

17 ,j.-^, 85

-16^^"^^ ==16'

5=./ 1-^

and

?V /^l_i?V- A' 12-'_ 13

1) ^\T) ^ V r'^'r ~T"'

/ 72 52Y /19 _ 62Y _ /395T

V V7 "^ 16/ "^ VT~ W ~ V "~Tl2

165^

22

^^Vl69 '^'

112" 112'

Hence

85 2 170 85 19 1615

13 -52__676 13 67 871

429 , 429

c^z = YY9,> and C2/3 = Yj2-

The coordinates of the centre of the incircle are therefore

170 _ 6^6 m 1615 871 429

1X2 ~ iT2 "^ 112 , IT2" "^ 112 '^ II2

85 13 429 85 13 429'

16 ^T"^ 112 16 "^y^ 112

-1 ,265

^ and ^.

The length of the radius of the incircle is the perpen-

dicular from ( â€” T^ , jYo ) ^P^^ *^Â® straight line

3aj + 42/ - 7 = 0,

THE STRAIGHT LINE. PROBLEMS. 77

and therefore ^

i^)^0

-21 + 1060-784 255 51

5x112 5x112 112'

The coordinates of the centre of the escribed circle

which touches the side BG externally are

_170_676 429 1615 871 429

112 112^112 ~Tl2" "^ 112"^Tl2

_85 13 429 85 13 429 '

"le'^T'^m ~l6'^T'^ri2

-417 , -315

-42~ ^""^ -^r-

Similarly the coordinates of the centres of the other

escribed circles can be written down.

100. Ex. Find the radius, and the coordinates of the centre, of

the circle circumscribing the triangle formed by the points

(0, 1), (2, 3), and (3, 5).

Let (iCi , 2/j) be the required centre and R the radius.

Since the distance of the centre from each of the three points is the

same, we have

^i'+ (2/1 - 1)'= (^1 - 2)H (^/i - 3)2= (.ri - 3)2+ (y, - 5)2= J22...(l).

From the first two we have, on reduction,

^1 + 2/1 = 3.

From the first and third equations we obtain

Solving, we have x^= - ^ and yi=^.

Substituting these values in (1) we get

i2=tx/10.

101. Ex. Prove that the middle points of the diagonals of a com-

plete quadrilateral lie on the same straight line.

[Complete quadrilateral. Def. Let OAGB be any quadrilateral.

Let AG and OB be produced to meet in E, and BG and OA to meet in

F. Join AB, OC, and EF. The resulting figure is called a complete

quadrilateral ; the lines AB,OG, and EF are called its diagonals, and

the points E, F, and D (the intersection of AB and OC) are called its

vertices.]

78 COOEDINATE GEOMETRY.

Take the lines OAF and OBE as the axes of x and y.

1Y

BI ^

< v

/n>

\c -.

â€¢.N

/ ^

* v^

n^X

[A

M-\

^^^

O A F X

Let 0A = 2a and 0B = 2b, so that A is the point (2a, 0) and B is

the point (0, 2b); also let C be the point {2h, 2k).

Then L, the middle point of OC, is the point {h, h), and ilf, the

middle point of AB, is (a, fo).

The equation to LM is therefore

Â£.e. {Ji-a)y-{k-h)x = bh-ak (1).

k-h

Again, the equation to BC is y -2b= - ~x.

Putting ?/ = 0, we have x = ^ â€” - , so that F is the point

f 2ak \

Similarly, E is the point ( 0, - _ j .

Hence N, the middle point of EF^ is f r; â€” r- , â– j .

These coordinates clearly satisfy (1), i.e. N lies on the straight

line L3I.

EXAMPLES. X.

1. A straight line is such that the algebraic sum of the perpen-

diculars let fall upon it from any number of fixed points is zero;

shew that it always passes through a fixed point.

2. Two fixed straight lines OX and Y are cut by a variable line

in the points A and B respectively and P and Q are the feet of the

perpendiculars drawn from A and B upon the lines OBY and OAX.

Shew that, if ^B pass through a fixed point, then PQ will also pass

through a fixed point.

[EXS. X.] THE STRAIGHT LINE. PROBLEMS. 79

3. If the equal sides AB and AC of au isosceles triangle be pro-

duced to E and F so that BE .GF = AB% shew that the line EF will

always pass through a fixed point.

4. If a straight line move so that the sum of the perpendiculars

let fall on it from the two fixed points (3, 4) and (7, 2) is equal to

three times the perpendicular on it from a third fixed point (1, 3),

prove that there is another fixed point through which this line always

passes and find its coordinates.

Find the centre and radius of the circle which is inscribed in the

triangle formed by the straight lines whose equations are

5. 3a; + 42/ + 2 = 0, 3x-4.y + 12 = 0, and 4x-3y = 0.

6. 2x + iy + S = 0, 4x + Sy + 3 = 0, and a;+l = 0.

7. y = 0, 12x-5y=0, and 3a; + 4?/-7 = 0.

8. Prove that the coordinates of the centre of the circle inscribed

in the triangle whose angular points are (1, 2), (2, 3), and (3, 1) are

8 + v/lQ and ^^-^^^Q

g ana g .

Find also the coordinates of the centres of the escribed circles.

9. Find the coordinates of the centres, and the radii, of the four

circles which touch the sides of the triangle the coordinates of whose

angular points are the points (6, 0), (0, 6), and (7, 7).

10. Find the position of the centre of the circle circumscribing

the triangle whose vertices are the points (2, 3), (3, 4), and (6, 8).

Find the area of the triangle formed by the straight lines whose

equations are

11. y = x, y = 2x, and y = Sx + 4.

12. y + x=0, y=x + G, and y = 7x + 5.

13. 2y + x-5 = 0, y + 2x-7 = 0, and x-y + l = 0.

14. Sx-'iy + 4a = 0, 2x-By + 4a = 0, and 5x-y + a = 0, proving also

that the feet of the perpendiculars from the origin upon them are

coUinear.

15. y = ax-bc, y = bx-ca, and y = cx-db.

16. y â€” m.x-\ â€” , y = in^-\ , and y = moX + - â€” .

17. y=m^x + Ci, y â€” m^x + c^, and the axis oiy.

18. y=v\x + c-^, y =m^ + c^, and y=m^ + c^.

19. Prove that the area of the triangle formed by the three straight

lines a^x + h^y + c^ = 0, a^x + K^y + Cg = 0., and a^x + Z>3i/ + Cg = is

1 ^

2 -

^ 2

- -^ K^2 - Â«2&l) (Â«2&3 - Â«3&2) (Â«3&1 " ^1^ â€¢

80 COORDINATE GEOMETRY. [ExS. X.]

20. Prove that the area of the triangle formed by the three straight

lines

X GOB a + y sin a- Pi = 0, xcos^ + y sin/S-^g^^?

and a; cos 7+?/ sin 7 -2)3 = 0,

^ sin (7 - /3) sin (a - 7) sin {^-a)

21. Prove that the area of the parallelogram contained by the

lines

4y-Sx-a=0, Sy -4:X + a = 0, 4:y-Sx-da=0,

and 3y-4:x + 2a=0 is faK

22. Prove that the area of the parallelogram whose sides are the

straight lines

aiX + biy + Ci = 0, ajX + bjy + dj^ = 0, a^x + b^y + c^^O,

and a^ + b2y + d2=0

IS

Â«1&2 - ^2^1

23. The vertices of a quadrilateral, taken in order, are the points

(0, 0), (4, 0), (6, 7), and (0, 3) ; find the coordinates of the point of

intersection of the two lines joining the middle points of opposite

sides.

24. The lines a; + 2/ + 1=0, x-y + 2=0, 4x + 2y + S=0, and

x + 2y-4: =

are the equations to the sides of a quadrilateral taken in order ; find

the equations to its three diagonals and the equation to the line on

points so determined lie on a straight line.

12. Find the coordinates of the point of intersection of the

straight lines

2x-3y=^l and 5y-x = S,

and determine also the angle at which they cut one another.

13. Find the angle between the two lines

Sx + y + 12 = and x + 2y-l = 0.

Find also the coordinates of their point of intersection and the

equations of lines drawn perpendicular to them from the point

(3, -2).

VIII.] EXAMPLES. 63

14. Prove that the points whose coordinates are respectively

(5, 1), (1, -1), and (11, 4) lie on a straight line, and find its intercepts

on the axes.

Prove that the following sets of three lines meet in a point.

15. 2x-Sy = 7, Sx-4:y = 13, and 8x-lly = S3.

16. dx + 4.y + G = 0, 6x + 5y + 9 = 0, and Sx + Sy + 5 = 0.

17. - + 7 = 1, j+^ = l, and y = x.

abba

18. Prove that the three straight lines whose equations are

15a;- 18?/ + 1 = 0, 12x + lOi/ - 3 = 0, and 6x + QQy-ll =

all meet in a point.

Shew also that the third line bisects the angle between the other

two.

19. Find the conditions that the straight lines

y = m-^x + ai, y = m^-\-a^, and y = m2X-\-a^

may meet in a point.

Find the coordinates of the orthocentre of the triangles whose

angular points are

20. (0,0), (2, -1), and (-1,3).

21. (1,0), (2,-4), and (-5,-2).

22. In any triangle ABG^ prove that

(1) the bisectors of the angles A, B, and C meet in a point,

(2) the medians, i.e. the lines joining each vertex to the middle

point of the opposite side, meet in a point,

and (3) the straight lines through the middle points of the sides

perpendicular to the sides meet in a point.

Find the equation to the straight line passing through

23. tlie point (3, 2) and the point of intersection of the lines

2x + Sy = l and Sx-Ay = Q.

24. the point (2, - 9) and the intersection of the lines

2x + 5y-8 = and 3x-4y=^S5.

25. the origin and the point of intersection of

x~y-4i=0 and lx + y + 20=0,

proving that it bisects the angle between them.

26. the origin and the point of intersection of the lines

X y ^ ^ X y ^

- + f = 1 and Y + ^ = 1.

a b b a

27. the point (a, b) and the intersection of the same two lines.

28. the intersection of the lines

x-2y-a=0 and x + 3y-2a =

64 COORDINATE GEOMETRY. [Exs.

and parallel to the straight line

29. the intersection of the lines

x + 2y + S = and 3x + iy + 7 =

and perpendicular to the straight line

y-x = 8.

30. the intersection of the lines

dx-iy + l = and 5x + y -1=0

and cutting off equal intercepts from the axes.

31. the intersection of the lines

2x~By = 10 and x + 2y:=Q

and the intersection of the lines

16a;-102/ = 33 and 12x + Uy + 29 = 0.

32. If through the angular points of a triangle straight lines be

drawn parallel to the sides, and if the intersections of these Hnes be

joined to the opposite angular points of the triangle, shew that the

joining lines so obtained will meet in a point.

33. Find the equations to the straight lines passing through the

point of intersection of the straight lines

Ax + By + C = and A'x + B'y + C'^0 and

(1) passing through the origin,

(2) parallel to the axis of y,

(3) cutting off a given distance a from the axis of y,

and (4) passing through a given point {x', y').

34. Prove that the diagonals of the parallelogram formed by the

four straight lines

^?>x + y = 0, ^?>y + x=^0, jBx + y = l, and JBy + x = l

are at right angles to one another.

35. Prove the same property for the parallelogram whose sides

are

- + 7=1, r + - = l, - + | = 2, and t + - = 2.

a b a a o a

36. One side of a square is inclined to the axis of x at an angle a

and one of its extremities is at the origin ; prove that the equations

to its diagonals are

y (cos a - sin a) = x (sin a + cos a)

and ?/ (sin a + cos a) + a: (cos a -sin a) = a.

Find the equations to the straight lines bisecting the angles

between the following pairs of straight lines, placing first the bisector

of the angle in which the origin lies.

37. x+ysJB = %-\-2JB and a;-?/ ^3 = 6-2^3.

VIII.] EXAMPLES. 65

38. 12x + 5y-4c = and Bx+4.tj + 7 = 0.

39. ix + Sij -7 = and 24^ + 7ij- 31 = 0.

40. 2x + y=4: and y + Sx = 5.

41. y-b=^ i>(^-Â«) and y-h = z ^(rc-a).

^ l-m2^ ' ^ 1-m'^^ '

Find the equations to the bisectors of the internal angles of the

triangles the equations of whose sides are respectively

42. 3x + 4:y=e, 12x-5y=B, and 4:X-3y + 12 = 0.

43. Sx + 5y=15, x + y=4:, and 2x + y = Q.

44. Find the equations to the straight lines passing through the

foot of the perpendLcular from the point {h, Jc) upon the straight line

Ax + By + G = and bisecting the angles between the perpendicular

and the given straight line.

45. Find the direction in which a straight Kne must be drawn

through the point (1, 2), so that its point of intersection with the line

x + y = 4: may be at a distance ^^6 from this point.

CHAPTER V.

THE STRAIGHT LINE {continued).

POLAR EQUATIONS. OBLIQUE COORDINATES.

MISCELLANEOUS PROBLEMS. LOCI.

88. To find the general equation to a straight line in

polar coordinates.

Let p be the length of the perpendicular Y from the

origin upon the straight line, and

let this perpendicular make an

angle a with the initial line.

Let P be any point on the

line and let its coordinates be r

and 6.

The equation required will

then be the relation between r, 6, p, and a.

From the triangle YP we have

p = r cos YOP = rcos{a-6)=^r cos (6 - a).

The required equation is therefore

r cos (6 â€” a) =p.

[On transforming to Cartesian coordinates this equation becomes

the equation of Art. 53.]

89. To find the polar equation of the straight line

joining the poiiits whose coordinates are (r^, 6^) and {r^, 6^.

THE STRAIGHT LINE. OBLIQUE COORDINATES. 67

Let A and B be the two given points and P any point

on the line joining them

whose coordinates are r and

e.

Then, since

AA0B-=AA0P+A POB,

we have

^.e.

I.e.

J r^r^ sin A OB = J r^r sin AOP + ^ ri\ sin POB,

r^r^ sin {6 2 â€” 6^ = r^r sin {0 â€” 6-^ + rr^ sin (0^ â€” 6),

sin (^,-^i) _ sin ((9-^1) sin ((92-^)

OBLIQUE COORDINATES.

90. In the previous chapter we took the axes to be

rectangular. In the great majority of cases rectangular

axes are employed, but in some cases oblique axes may be

used with advantage.

In the following articles we shall consider the proposi-

tions in which the results for oblique axes are different

from those for rectangular axes. The propositions of Arts.

50 and 62 are true for oblique, as well as rectangular,

coordinates.

91. To find the equation to a straight line referred to

axes inclined at an angle w.

Let LPL' be a straight line which cuts the axis of

a distance c from the origin and is

inclined at an angle to the axis

of X.

Let P be any point on the

straight line. Draw PNM parallel

to the axis of y to meet OX in M^

and let it meet the . straight line

through C parallel to the axis of x

in the point N,

Let P be the point (.r, y\ so that

CN^OM^x, and NP = MP- 00 ^y-c.

5-

Fat

68 COORDINATE GEOMETRY.

Since L GPN= l FNN' - l PCN' = w - ^, we have

y-c NP _ B>inNC P _ sinO

~ir ~ 'CN~ ^^PN~ sin (o)- ^) â€¢

TT Si^^ /1\

Hence y = x-. â€” ; 7:. + c (1).

^ sm(o>-^) ^ ^

3?his equation is of the form

y = mx + c,

where

sin^ sin^ tan^

7)1 =

sin (o) â€” 6) sin w cos â€” cos to sin sin <o â€” cos w tan ^ '

, , â€ž J /> ^ sin CD

and thererore tan v = .

1 + 7)1 cos a>

In oblique coordinates the equation

y = mx + c

therefore represents a straight line which is inclined at an

angle

- m sin o)

tan~i

l+mcosco

to the axis of x.

Cor. From. (1), by putting in succession equal to 90Â°

and 90Â° + o), we see that the equations to the straight lines,

passing through the origin and perpendicular to the axes of

X and y, are respectively y = and y = â€”x cos w.

92. The axes being oblique, to find the equation to the

straight line, such that the 'perpendicular on it from the origin

is of length p and makes angles a and ^ with the axes of x

and y.

Let LM be the given straight line and OK the perpen-

dicular on it from the origin.

Let P be any point on the

straight line ; draw the ordinate

PN and draw NP perpendicular

to OK and PS "oerpendicular to

NR.

Let P be the point {x, y), so

that OJV = X and NP = y.

THE STRAIGHT LINE. OBLIQUE COORDINATES. 69

The lines NP and Y are parallel.

Also OK and SP are parallel, each being perpendicular

to NB.

Thus lSPN^lKOM=^.

We therefore have

'p = OK- OR + SP = OiVcos a + NP cos p=^xcosa + y cos ^.

Hence x cos a + y cos /5 â€” p = 0,

being the relation which holds between the coordinates of

any point on the straight line, is the required equation.

93. To find the angle between the straight lines

y = mx + c and y = mx + c,

the axes being oblique.

If these straight lines be respectively inclined at angles

and 0' to the axis of x, we have, by the last article,

^ msinoD , ^ ,,, m'sinw

tan u = :. and tan u â€”

1 + ni cos CO 1 + 711 cos 0)

The angle required is 0~ 0'.

XT J. ir\ t\t\ tan ^- tan ^'

Now tan(e-e)=j^:^^^^^-^^,

m sm CD in sm co

1 +m cos CO 1 + 771 cos CO

m sin CO m' sin co

1 +

1+771 cos CO 1 + m' cos CO

_ m sin CO ( 1 + m' cos co) â€” m' sin co (1 + 7?i. cos co)

(1 + m cos co) (1 + W cos co) + ?92m' sin^ co

_ (m â€” w') sin CO

1 + (m + m ) cos CO + mm' '

The required angle is therefore

tan

_j (m â€” in!) sin co

1 + {m + TTb) cos 0) + 7f}im' '

Cor. 1. The two given lines are parallel if m = m'.

Cor. 2. The two given lines are perpendicular if

1 + (m + m') cos 0} + mm' = O.

70 COORDINATE GEOMETRY.

94. If the straight lines have their equations in the

form

Ax-\-By + G = and A'x + B'y â– \- C = 0,

then 7n = - ^ and m = - Wt'

Substituting these values in the result of the last article

the angle between the two lines is easily found to be

J A'B-AB' .

AA' + BB' - {AB' + A'B) cos w

The given lines are therefore parallel if

A'B-AB'=^0.

They are perpendicular if

AA' + BB' = {AB' + A'B) cos w.

95. Ex. The axes being inclined at an angle of 30Â°, obtain the

equations to the straight lines ivhich pass through the origin and are

inclined at 45Â° to the straight line x + y = l.

Let either of the required straight lines be y = mx.

The given straight line isy= -x + 1, so that m'= - 1.

We therefore have

1 + (m + m') cos <a + mm'

where m'= - 1 and w = 30Â°.

rw,- . â– â– â– m+1 . -

This equation gives 2 + (,,,_ 1)^3- 2m = "^^^

Taking the upper sign we obtain m= â€” j^.

Taking the lower sign we have m = - ^3.

The required equations are therefore

y=-sjSx and y=z - ^x,

i.e. y + iJ3x=^0 and JSy + x = 0.

96. To find the length of the lyerpendicular froin the

point (x'j y) upon the straight line Ax + By +0 â€” 0, the axes

being inclined at an angle <a, and the equation being written

so that C is a negative quantity.

THE STRAIGHT LINE. OBLIQUE COORDINATES. 71

Let the given straight line meet the axes in L and M^

G C

so that OL = - , and OM^ â€” -=, .

A B

Let P be the given point {x', y').

Draw the perpendiculars PQ, PR,

and PS on the given line and the

two axes.

Taking and P on opposite sides

of the given line, we then have

IiLPM + AMOL=aOLP + aOPM,

i.e. PQ . LM + OL . OM silicon OL . PE + OM . PS. ..{I).

Draw PU and PV parallel to the axes of y and x, so

that PU = y' and PV-^x.

Hence PE ^ PU sin PUR = y' sin w,

and PS =PV sin P VS - x sin w.

Also

LM= s/OL^ + OM^ - 20 L . Oif cos w

V

(72 C'

0^

z^-'^-^zg'"'"^

- (7 /l+l

2 cos w

since C is a negative quantity.

On substituting these values in (1), we have

pc><(-o)xy-i+-i

2 cos CO C^ .

c

c

so that

PQ =

- 7.7/ sm o> â€” Pi . Â£c sm 00,

Ax^ + By' + C

VA2 + B2 - 2 AB cos CO

. sm a;.

Cor. If (0 â€” 90Â°, i.e. if the axes be rectangular, we

have the result of Art. 75.

72 COORDINATE GEOMETRY.

EXAMPLES. IX.

1. The axes being inclined at an angle of 60Â°, find the inclination

to the axis of x of the straight lines whose equations are

(1) 2/=2^ + 5,

and (2) 2y=.{^^-l)x + l.

2. The axes being inclined at an angle of 120Â°, find the tangent

of the angle between the two straight lines

Qx + ly=.l and 28a; - 73?/ = 101.

3. With oblique coordinates find the tangent of the angle

between the straight lines

y = mx + c and my+x = d.

4. liy=x tan -â€” - and y=x tan â€” j represent two straight lines

at right angles, prove that the angle between the axes is ~ .

5. Prove that the straight lines y + x=c and y=x + d are at

right angles, whatever be the angle between the axes.

6. Prove that the equation to the straight line which passes

through the point {h, Jc) and is perpendicular to the axis of x is

x + y cos o) = h+k cos w.

7. Find the equations to the sides and diagonals of a regular

hexagon, two of its sides, which meet in a corner, being the axes of

coordinates.

8. From each corner of a parallelogram a perpendicular is drawn

upon the diagonal which does not pass through that corner and these

are produced to form another parallelogram ; shew that its diagonals

are perpendicular to the sides of the first parallelogram and that they

both have the same centre.

9. If the straight lines y = miX + Cj^ and y=m,^x + C2 make equal

angles with the axis of x and be not pariallel to one another, prove

that iiij^ + ^2 + 2mjin2 cos w = 0.

10. The axes being inclined at an angle of 30Â°, find the equation

to the straight line which passes through the point ( - 2, 3) and is

perpendicular to the straight line y + Bx = 6.

11. Find the length of the perpendicular drawn from the point

(4, -3) upon the straight line 6a; + 3^ -10 = 0, the angle between the

axes being 60Â°.

12. Find the equation to, and the length of, the perpendicular

drawn from the point (1, 1) upon the straight line 3a; + 4?/ + 5 = 0, the

angle between the axes being 120Â°.

[EXS. IX.] THE STRAIGHT LINE. PROBLEMS. 73

13. The coordinates of a point P referred to axes meeting at an

angle w are [h, k) ; prove that the length of the straight line joining

the feet of the perpendiculars from P upon the axes is

sin w ^y/i^ +k- + 2hk cos w.

14. From a given point {h, k) perpendiculars are drawn to the

axes, whose inclination is oj, and their feet are joined. Prove that

the length of the perpendicular drawn from [h, Jc) upon this line is

hk sin^ 0}

JhF+W+2hkcos^'

and that its equation is hx - ky = h^- k^.

Straight lines passing through fixed points.

97. If the equation to a straic/ht line be of the form

ax + hy + c + \ (ax + b'y + g') = (1 ),

where \ is any arbitrary constant^ it always passes through

one fixed point whatever be the value of \.

For the equation (1) is satisfied by the coordinates of

the point which satisfies both of the equations

ax + by + G~Oj

and a'x 4- b'y + c' = 0.

This point is, by Art. 77,

'be' â€” b'c ca' â€” c'a^

^ab' â€” a'b ' ab' â€” a'b/

and these coordinates are independent of A.

Ex. Given the vertical angle of a triangle in magnitude and

position, and also the sum of the reciprocals of the sides ivhich contain

it; shew that the base always passes through a fixed point.

Take the fixed angular point as origin and the directions of the

sides containing it as axes ; let the lengths of these sides in any such

triangle be a and &, which are not therefore given.

We have - + 7=const. = r- (say) (1).

do K

The equation to the base is

^ y -r

â€” v- -=1

a

z...,by(l), ^'+^(1_1)^1,

1 y

i.e. -(x-y) + ^-l = 0.

74

COORDINATE GEOMETRY.

"Whatever be the value of a this straight line always passes through

the point given by

x-y = and |-1 = 0,

i.e. through ih.e fixed point {k, k).

k

98. Prove that the coordinates of the centre of the

circle inscribed in the triangle, whose vertices are the points

(^ij 2/i)> (^2 J 2/2)5 Â«^^G? (a^3, 2/3), are

axi + hx^ + cx^ ay^ + hy.2 + cy^

a+b+G a+b+c '

where a, 6, Â«7ic? c are ^Ae lengths of the sides of the triangle.

Find also the coordinates of the centres of the escribed

circles.

Let ABC be the triangle and let AD and CE be the

bisectors of the angles A and G

and let them meet in 0'.

Then 0' is the required point.

Since AD bisects the angle

BAC we have, by Euc. YI. 3,

^ _DG_ BD + DG _ _a_

BA ~ AG~ BA+'AC~bTc'

so that

ba

(xj.yj)

'(-,,y,)

DG

b + c

Also, since GO' bisects the angle AGD, we have

^' _ JLC _ _5_ _ 6 + c

0'D~ CD~ ba

a

b + <

The point D therefore divides BG in the ratio

BA : AG, i.e. c : b.

Also 0' divides AD in the ratio b + c : a.

Hence, by Art. 22, the coordinates of D are

cxs + bx^ fi ^3 + ^.'/2

c + b

c + b

THE STRAIGHT LINE. PROBLEMS.

Also, by the same article, the coordinates of 0' are

cxo + hxo ,, X cVo + hy.^

(b + c) X â€”^ â€” â€”^ + ax, (b + c)x -^ â€” ^^ + ay.

^ ' G+h T ' G+b ^

and

{b + c) + a (b + c) +

a

ax^ + bx^, + Gx.^ - ay^ 4- by^ + cy^

a + b + G a + b + c

Again, if 0^ be the centre of the escribed circle opposite

to the angle -4, the line COi bisects the exterior angle of

ACB.

Hence (Euc. VI. A) we have

AO, _AC__ b^G

Therefore Oi is the point which divides AD externally in

the ratio b + g : a.

Its coordinates (Art. 22) are therefore

,y . GXo + bx^ ,-, , cy., + byÂ»

(b + c) ^ ^ ^ - ax^ (b + c) -^ â€” ^ - ay,

^ ^ G + b ^ , ^ ' c + b ^^

ana

{b + G) â€” a (^ + c)

a

â€” axi + bx^ + Gx^ - â€” ayi + by. 2 4- Gy^

â€”a+b+G â€”a+b+c

Similarly, it may be shewn that the coordinates of the

escribed circles opposite to B and C are respectively

^aXj^ â€” bx2 + GXs ay^ â€” by^ + cyA

c-

aâ€”b+G ' aâ€”b+c

and /axj^x2-_cxs ay^ + by.^ - Gy ^

\ a + b â€” c ' a + b â€” G

99. As a numerical example consider the case of the

triangle formed by the straight lines

3x+4cy-7 = 0, 12x + 6y-l7 =0 and 5x + 12y - 34: = 0.

These three straight lines being BC, CA, and AB

respectively we easily obtain, by solving, that the points

A, B, and C are

Q' t)' (t?' 11) ^"'^ (i'^>-

76 COORDINATE GEOMETRY,

Hence

yfi?-')"*(s-')'V

682 5p

17 ,j.-^, 85

-16^^"^^ ==16'

5=./ 1-^

and

?V /^l_i?V- A' 12-'_ 13

1) ^\T) ^ V r'^'r ~T"'

/ 72 52Y /19 _ 62Y _ /395T

V V7 "^ 16/ "^ VT~ W ~ V "~Tl2

165^

22

^^Vl69 '^'

112" 112'

Hence

85 2 170 85 19 1615

13 -52__676 13 67 871

429 , 429

c^z = YY9,> and C2/3 = Yj2-

The coordinates of the centre of the incircle are therefore

170 _ 6^6 m 1615 871 429

1X2 ~ iT2 "^ 112 , IT2" "^ 112 '^ II2

85 13 429 85 13 429'

16 ^T"^ 112 16 "^y^ 112

-1 ,265

^ and ^.

The length of the radius of the incircle is the perpen-

dicular from ( â€” T^ , jYo ) ^P^^ *^Â® straight line

3aj + 42/ - 7 = 0,

THE STRAIGHT LINE. PROBLEMS. 77

and therefore ^

i^)^0

-21 + 1060-784 255 51

5x112 5x112 112'

The coordinates of the centre of the escribed circle

which touches the side BG externally are

_170_676 429 1615 871 429

112 112^112 ~Tl2" "^ 112"^Tl2

_85 13 429 85 13 429 '

"le'^T'^m ~l6'^T'^ri2

-417 , -315

-42~ ^""^ -^r-

Similarly the coordinates of the centres of the other

escribed circles can be written down.

100. Ex. Find the radius, and the coordinates of the centre, of

the circle circumscribing the triangle formed by the points

(0, 1), (2, 3), and (3, 5).

Let (iCi , 2/j) be the required centre and R the radius.

Since the distance of the centre from each of the three points is the

same, we have

^i'+ (2/1 - 1)'= (^1 - 2)H (^/i - 3)2= (.ri - 3)2+ (y, - 5)2= J22...(l).

From the first two we have, on reduction,

^1 + 2/1 = 3.

From the first and third equations we obtain

Solving, we have x^= - ^ and yi=^.

Substituting these values in (1) we get

i2=tx/10.

101. Ex. Prove that the middle points of the diagonals of a com-

plete quadrilateral lie on the same straight line.

[Complete quadrilateral. Def. Let OAGB be any quadrilateral.

Let AG and OB be produced to meet in E, and BG and OA to meet in

F. Join AB, OC, and EF. The resulting figure is called a complete

quadrilateral ; the lines AB,OG, and EF are called its diagonals, and

the points E, F, and D (the intersection of AB and OC) are called its

vertices.]

78 COOEDINATE GEOMETRY.

Take the lines OAF and OBE as the axes of x and y.

1Y

BI ^

< v

/n>

\c -.

â€¢.N

/ ^

* v^

n^X

[A

M-\

^^^

O A F X

Let 0A = 2a and 0B = 2b, so that A is the point (2a, 0) and B is

the point (0, 2b); also let C be the point {2h, 2k).

Then L, the middle point of OC, is the point {h, h), and ilf, the

middle point of AB, is (a, fo).

The equation to LM is therefore

Â£.e. {Ji-a)y-{k-h)x = bh-ak (1).

k-h

Again, the equation to BC is y -2b= - ~x.

Putting ?/ = 0, we have x = ^ â€” - , so that F is the point

f 2ak \

Similarly, E is the point ( 0, - _ j .

Hence N, the middle point of EF^ is f r; â€” r- , â– j .

These coordinates clearly satisfy (1), i.e. N lies on the straight

line L3I.

EXAMPLES. X.

1. A straight line is such that the algebraic sum of the perpen-

diculars let fall upon it from any number of fixed points is zero;

shew that it always passes through a fixed point.

2. Two fixed straight lines OX and Y are cut by a variable line

in the points A and B respectively and P and Q are the feet of the

perpendiculars drawn from A and B upon the lines OBY and OAX.

Shew that, if ^B pass through a fixed point, then PQ will also pass

through a fixed point.

[EXS. X.] THE STRAIGHT LINE. PROBLEMS. 79

3. If the equal sides AB and AC of au isosceles triangle be pro-

duced to E and F so that BE .GF = AB% shew that the line EF will

always pass through a fixed point.

4. If a straight line move so that the sum of the perpendiculars

let fall on it from the two fixed points (3, 4) and (7, 2) is equal to

three times the perpendicular on it from a third fixed point (1, 3),

prove that there is another fixed point through which this line always

passes and find its coordinates.

Find the centre and radius of the circle which is inscribed in the

triangle formed by the straight lines whose equations are

5. 3a; + 42/ + 2 = 0, 3x-4.y + 12 = 0, and 4x-3y = 0.

6. 2x + iy + S = 0, 4x + Sy + 3 = 0, and a;+l = 0.

7. y = 0, 12x-5y=0, and 3a; + 4?/-7 = 0.

8. Prove that the coordinates of the centre of the circle inscribed

in the triangle whose angular points are (1, 2), (2, 3), and (3, 1) are

8 + v/lQ and ^^-^^^Q

g ana g .

Find also the coordinates of the centres of the escribed circles.

9. Find the coordinates of the centres, and the radii, of the four

circles which touch the sides of the triangle the coordinates of whose

angular points are the points (6, 0), (0, 6), and (7, 7).

10. Find the position of the centre of the circle circumscribing

the triangle whose vertices are the points (2, 3), (3, 4), and (6, 8).

Find the area of the triangle formed by the straight lines whose

equations are

11. y = x, y = 2x, and y = Sx + 4.

12. y + x=0, y=x + G, and y = 7x + 5.

13. 2y + x-5 = 0, y + 2x-7 = 0, and x-y + l = 0.

14. Sx-'iy + 4a = 0, 2x-By + 4a = 0, and 5x-y + a = 0, proving also

that the feet of the perpendiculars from the origin upon them are

coUinear.

15. y = ax-bc, y = bx-ca, and y = cx-db.

16. y â€” m.x-\ â€” , y = in^-\ , and y = moX + - â€” .

17. y=m^x + Ci, y â€” m^x + c^, and the axis oiy.

18. y=v\x + c-^, y =m^ + c^, and y=m^ + c^.

19. Prove that the area of the triangle formed by the three straight

lines a^x + h^y + c^ = 0, a^x + K^y + Cg = 0., and a^x + Z>3i/ + Cg = is

1 ^

2 -

^ 2

- -^ K^2 - Â«2&l) (Â«2&3 - Â«3&2) (Â«3&1 " ^1^ â€¢

80 COORDINATE GEOMETRY. [ExS. X.]

20. Prove that the area of the triangle formed by the three straight

lines

X GOB a + y sin a- Pi = 0, xcos^ + y sin/S-^g^^?

and a; cos 7+?/ sin 7 -2)3 = 0,

^ sin (7 - /3) sin (a - 7) sin {^-a)

21. Prove that the area of the parallelogram contained by the

lines

4y-Sx-a=0, Sy -4:X + a = 0, 4:y-Sx-da=0,

and 3y-4:x + 2a=0 is faK

22. Prove that the area of the parallelogram whose sides are the

straight lines

aiX + biy + Ci = 0, ajX + bjy + dj^ = 0, a^x + b^y + c^^O,

and a^ + b2y + d2=0

IS

Â«1&2 - ^2^1

23. The vertices of a quadrilateral, taken in order, are the points

(0, 0), (4, 0), (6, 7), and (0, 3) ; find the coordinates of the point of

intersection of the two lines joining the middle points of opposite

sides.

24. The lines a; + 2/ + 1=0, x-y + 2=0, 4x + 2y + S=0, and

x + 2y-4: =

are the equations to the sides of a quadrilateral taken in order ; find

the equations to its three diagonals and the equation to the line on

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