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S. L. (Sidney Luxton) Loney.

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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>(^-«) 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 — 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 — 6^ = r^r sin {0 — 6-^ + rr^ sin (0^ — 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)- ^) •

TT Si^^ /1\

Hence y = x-. — ; 7:. + c (1).

^ sm(o>-^) ^ ^

3?his equation is of the form

y = mx + c,
where

sin^ sin^ tan^



7)1 =



sin (o) — 6) sin w cos — cos to sin sin <o — cos w tan ^ '

, , „ 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 90°
and 90° + 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 = —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 — 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 —



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) — m' sin co (1 + 7?i. cos co)

(1 + m cos co) (1 + W cos co) + ?92m' sin^ co
_ (m — w') sin CO

1 + (m + m ) cos CO + mm' '
The required angle is therefore



tan



_j (m — 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 ■\- 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 30°, obtain the
equations to the straight lines ivhich pass through the origin and are
inclined at 45° 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 = 30°.

rw,- . ■■ ■ m+1 . -

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

Taking the upper sign we obtain m= — 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 — 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^ — -=, .
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> — Pi . £c sm 00,

Ax^ + By' + C

VA2 + B2 - 2 AB cos CO



. sm a;.



Cor. If (0 — 90°, 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 60°, 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 120°, 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 -— - and y=x tan — 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 30°, 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 60°.

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 120°.



[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' — b'c ca' — c'a^



^ab' — a'b ' ab' — 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

— 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 «^^G? (a^3, 2/3), are

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

a+b+G a+b+c '

where a, 6, «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 —^ — —^ + ax, (b + c)x -^ — ^^ + 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., + by»

(b + c) ^ ^ ^ - ax^ (b + c) -^ — ^ - ay,

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

ana



{b + G) — a (^ + c)



a



— axi + bx^ + Gx^ - — ayi + by. 2 4- Gy^
—a+b+G —a+b+c

Similarly, it may be shewn that the coordinates of the
escribed circles opposite to B and C are respectively

^aXj^ — bx2 + GXs ay^ — by^ + cyA



c-



a—b+G ' a—b+c



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

\ a + b — c ' a + b — 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 ( — T^ , jYo ) ^P^^ *^® 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 -.


•.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

£.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 = ^ — - , 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; — r- , ■ 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 — m.x-\ — , y = in^-\ , and y = moX + - — .

17. y=m^x + Ci, y — 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 - «2&l) («2&3 - «3&2) («3&1 " ^1^ •



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



«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



Online LibraryS. L. (Sidney Luxton) LoneyThe elements of coordinate geometry → online text (page 5 of 26)