Two Dimensional Co-Ordinate Geometry

7/28/2019 Two Dimensional Co-Ordinate Geometry

http://slidepdf.com/reader/full/two-dimensional-co-ordinate-geometry 1/29

Two Dimensional Co-ordinate Geometry Advanced Level Pure Mathematics

Advanced Level Pure Mathematics

Calculus II

9.1 Introduction

9.2 Change of Axes

9.3 Straight Lines

9.4 Equations of Lines Pairs

9.5 Circle

9.6 Parabola

9.7 Ellipse

9.8 Hyperbola

9.1 Introduction

• If 21 m,m be the gradients of two straight lines respectively, angle θ between them,

the θ is given by

Prepared by K. F. Ngai

9




tan θ =21

21

mm1

mm

+

−

• If the points 21 P,P and P are collinear, then P is said to divide the line segment

21PP in the ratio 21m:m or r where r = .

PP

PP

2

1

We have

x = ,r 1

rxx 21

++

y =r 1

ryy 21

++

If P divides 21PP internally, r is positive.

If P divides 21PP externally, r is negative.

• Area of triangle is

1yx

1yx

1yx

2

1

33

22

11

counter-clockwise

Hence area of quadrilateral vertices are A: ),y,x( ii i = 1, 2, 3, 4 arranged in

counter-clockwise is

+++

11

44

44

33

33

22

22

11

yx

yx

yx

yx

yx

yx

yx

yx

2

1

• Area of n-sided polygon is

++++ −−

11

nn

nn

1n1n

33

22

22

11

yx

yx

yx

yx...

yx

yx

yx

yx

2

1

Condition for collinearity of 3 points is

1yx

1yx1yx

33

22

11

= 0

• Parametric Equation


P1 (x1, y1)

P2 (x2, y2)

P (x, y)

r

1

:




Given a pair of equation x = x(t), y = y(t), when we eliminate the variable t, we

obtain equation f(x, y) = 0.

Example 1 F: axy3yx 33 −+ = 0, a > 0

By considering the intersection of line y = tx and curve F. Show curve F

may represented parametrically by

x = ,t1

at32+

y =2

2

t1

at3

+

9.2Change of Axes1. Translation of axes

Take new axes O′X′ and O′Y′ parallel

to OX and OY where O′(h, k).

Let the old and

new co-ordinates

of P be (x, y) and (x′, y′) we have

x = x′ + h

y = y′ + k


y ′

x′

x

0′ (h, k)

× P




2. Rotation of co-ordinates axes

Let the old and new coordinates of P be (x, y) and (X, Y)

x = OP cos (θ + φ)

= OP cos θ cos φ − OP sin θ sin φ

= X cos θ + Y sin θ

y = OP sin (θ + φ)

= OP sin θ cos φ − OP cos θ sin φ

= X sin θ + Y cos θ

we have

y

x=

θθθ−θ

Y

X

cossin

sincos

(old) (new)

Y

X=

θθ−θθ

y

x

cossin

sincos

(new) (old)

Example 2 Give E: 31y18x16y3x4 22 ++−+ = 0 if the origin of co-ordinates system is

translated to (2, −3), find the equation of curve in new coordinates system.


y

y

x

xθ

φ

P




Example 3 Let C be a curve in Ox-y plane with equationy6x12y4xy4x 22 −+++ = 0. If the axes are rotated through an angle

θ = ,2tan 1− find equation C in new coordinates system.

Example 4 (a) The complex numbers z = x + iy, w = u + vi, x, y, u, v ∈ R, are related

by the equation w = .

z

12

Show u = 222

22

)yx(

yx

+

−and v = 222

)yx(

xy2

+

−

(b) One family of curves 1F in x-y plane is given by u = λ and another

family 2F is given by u = µ. λ, µ are parameters. Show that at each

point of ,F1 dx

dy=

)yx3(y

)xy3(x22

22

−

−

(c) Show that each curve of 1F at intersect every curve of 2F at right

angles.





9.3 Straight Lines

• Point slope form: y − 1y = )xx(m1

−

Parametric equation: x = 1x + t cos θ

y = 1y + t sin θ

t ∈ (−∞, ∞) θ = mtan 1−

• Slope-intercept form:y = mx + c

• Two-point form

1

1

yy

xx

−

−=

21

21

yy

xx

−

−

Parametric form: x = 21x)t1(tx −+

y = 21y)t1(ty −+

Proof) x = ,r 1

rxx21

+

+y =

r 1

ryy21

+

+

∴ x = 21xr 1

1xr 1

1

++

+

⇒ x = ,x)t1(tx21

−+ set t =r 1

1

+

Similarly,

y = 21 yr 1

1yr 1

1

+++

⇒ y = 21y)t1(ty −+

• Intercept form: b

y

a

x+ = 1

Example 5 The line CD makes interceptsh

a2

andk

b2

on x and y-axis respectively.




y

x

p


(a) Find the coordinates Q, the point of intersection of line b

y

a

x+ = 1 and CD.

(b) Prove that if the point (h, k) lies on b

y

a

x+ = 1, then the equation of line joining

Q to the origin is kx + hy = 0.

• Normal form: x cos α + y sin α = p

Given Ax + By + C = 0

Normal form:

222222BA

cy

BA

Bx

BA

A

+±+

+±+

+±= 0

The sign is taken as follow:

1. If C ≠ 0, the sign is chosen as opposite to that of C. It is because the constant

term in normal form is always negative.

2. If C = 0, the sign is chosen as the same as that of B.

Example 6 The line 1L is given by the equation 3x − y − 2 = 0. Find the equation of


α




(a) Line 2L which passes through the point (1, 1) and is ⊥ 1L

Prove that the parametric equation of 1L and 2L are respectively

:L1 x = 1 + ,t1 y = 1 + ;t31

:L2 x = 1 − ,t3 2 y = 1 + 2t

(b) The points P on 1L and Q on 2L vary so that PQ is always 3 units in

length, R is the point on PQ such that PR = 2RQ. Prove that the locus

of R is 22 )4y3x(4)2yx3( −++−− = 40.

Example 7 (a) Given a straight line L: ax + by + c = 0.

Find the mirror image of the point )y,x(P 11 in the line L. Hence findthe mirror image of the line :L1 x + my + n = 0 in the line L.

(b) Find the equation of locus of the image of a variable point (cos θ, sin θ)





where 0 ≤ θ < 2π, in the line x + y = .2

• Angle between two straight lines

tan θ =21

21

mm1

mm

+

−





• Distance of a point from a line

i. Given L: x cos α + y sin α − p = 0 and point )y,x( 00

d = psinycosx 00 −α+α

Proof) Draw a line // L and pass through point )y,x( 00

)d p(sinycosx:L0 +−α+α = 0

pass through )y,x( 00

∴ )d p(sinycosx 00 +−α+α = 0

⇒ d = psinycosx 00 −α+α

If )y,x( 00 and the origin are on the opposite side of line L, the distance d is

positive; If they are on the same side of the line L, d is negative.

ii. Given: Ax + By + C = 0 and )y,x( 00

d =22

00

BA

CByAx

+

++

• The equation of angle bisectors

Thm. The equation to the bisectors of the angles between two straight lines


(x0, y0)

y

x

L

L0

d

d

p




111CyBxA ++ = 0 and 222

CyBxA ++ = 0 are

2

1

2

1

111

BA

CyBxA

+

++= 2

2

2

2

222

BA

CyBxA

+

++

Proof) Let P = (x, y) be any point on the angle bisectors of the given lines, then P will

be equidistant from both lines. Hence the coordinates (x, y) of P will satisfy

2

1

2

1

111

BA

CByAx

+

++=

2

2

2

2

222

BA

CByAx

+

++

or 2

1

2

1

111

BA

CyBxA

+

++= 2

2

2

2

222

BA

CyBxA

+

++±

which are the required equations.

Example 8 Two perpendicular lines are drawn through the origin so as to form an

isosceles right-angled triangle with the line

x + my + n = 0.

Show that their equations are

( − m)x + ( + m)y = 0 and (+ m)x − ( − m)y = 0.

• Condition for concurrency

Thm. Three lines not all parallel represented by

iiiiCyBxA:L ++ = 0 i = 1, 2, 3





are concurrent iff

333

222

111

CBA

CBA

CBA

= 0.

Example 9 The equations

=+λ−

=+−

=−λ+−λ

02yx4

01y2x3

03y3x2

represent three straight lines in the x-y

plane.

(a) Find the values of λ for which the lines are concurrent.

(b) For each of these values λ, find the coordinates of the point at which

the lines are concurrent.

• System of straight lines

The equation )CyBxA(111

++ + )CyBxA(222

++λ = 0,

λ: arbitrary constant, represents a straight lines passing through the intersection of

the lines )CyBxA( 111 ++ = 0 and 222 CyBxA ++ = 0.

Example 10 The equations of four lines EAB, BCF, CDE, FDA are 3x − 2y + 1 = 0,

4x − y + 2 = 0, 2x + y + 2 = 0 and





2(3x − 2y + 1) − 3(4x − y + 2) − (2x + y + 2) = 0. Obtain, without finding

the coordinates of B and D, the equation of straight line BD.

9.4 Equations of Lines PairsConsider the pair of straight lines through origin y bxa 11 + =0

y bxa22

+ = 0

The combined equation of line pairs is

)y bxa)(y bxa( 2211 ++ = 0 or 2

211221

2

21y b bxy) ba ba(xaa +++ = 0.

Remark:

The pair of straight lines22

byhxy2ax ++ = 0 has the following properties:1. real and distinct if .abh 2 >

2. real and coincident if .abh 2 ≥

3. imaginary if .abh 2 <

Example 11 L: qx + py = pq be a straight line and M: 22 byhxy2ax ++ = 0 a line pair

through origin.

(a) If L meets M at A )y,x( 11 and B ),y,x( 22 Show

21 xx + = 22 bqhpq2ap

)hp bq( pq2

+−

−

21xx = 22

22

bqhpq2ap

q bp

+−

(b) (i) Prove

2)BA( ⋅ = 222

22222

] bqhpq2ap[

)abh)(q p(q p4

+−

−+

(ii) If OA = OB, show

)q p(h 22 − = pq(b − a)





Example 12 Express in a single equation of perpendicular

(a) straight lines through the origin, one of which is the line ax + by = 0.

(b) Find also a single equation of the pair of angle bisection of the line pair

in (a).





Thm. The condition that the general equation of the second degree in x and y

Γ : cfy2gx2 byhxy2ax 22 +++++ = 0

shall represent two straight lines is

cf g

f bh

gha

= 0.

Proof) The given equation can be written as

)cfy2 by()ghy(x2ax22 +++++ = 0

Solving this equation as a quadratic in x,

x =a2

)cfy2 by(a4)ghy(4)ghy(222 ++−+±+−

= a2

acafy2abyghgy2yh)ghy(222 +−−++±+−

ax + (hy + g) = )acg()af hg(y2)abh(y 222 −+−+−±

In order that Γ shall represent two straight lines, the left hand side of Γ should be written

as a product of two linear factors of the form Ax + By + C = 0, the quantity under the

root sign in the above equation must be a complete square.

Thus the condition is)acg)(abh()af hg( 222 −−−− = 0

which reduces to222 ch bgaf fgh2abc −−−+ = 0

This result can be put in the determinant form

cf g

f bh

gha

= 0

Example 13 Let Γ : 13x8y10xxy4y 22 ++−+− = 0

(a) Prove Γ represents a pair of straight lines. Find the separate equations

of the line pair.

(b) Find their point of intersection and the angle between them.





Example 14 (a) Show that the bisectors of the angles between the lines22 byhxy2ax ++ = 0 are given by the equation

22hyxy) ba(hx −−− = 0

(b) Show that the equation

5y12x14y4xy4x3 22 −++−− = 0

represents two lines, and find(i) the coordinates of the intersection of the line pair, and

(ii) the combined equation of the bisectors of the angles between

them.




(x, y)

(x1, y1) (x2, y2)

(x, y)


9.5 Circle1. 22 yx + = 2r represents a circle, centre (0, 0).

2. Centre (h, k) 22)k y()hx( −+− = 2r

3. In second degree homogenous equation

cfy2gx2 byhxy2ax 22 +++++ = 0

Condition for circle = a = b, h = 0

centre )a

f ,

a

g( −−

radius =a

c

a

f

a

g2

2

2

2

−+

Circle: cfy2gx2yx 22 ++++ = 0

4. Circle which has diameter with end points )y,x(),y,x( 2211 are

−−

−−

2

2

2

1

xxyy

xxyy = −1

5. Equation of tangent at point )y,x( 11 are

c)yy(f )xx(gyyxx1111

++++++ = 0





6. Equation of tangent with slope mIf circle equation 22 yx + = 2

r

Let equation of tangent is y = mx + c,

⇒ 22)cmx(x ++ = 2

r

∴ c = 2m1r +±

∴ y = mx 2m1r +±

7. Let P )y,x( 11 be a point lying outside the circle cfy2gx2yx 22 ++++ = 0, the equation

of chord is

c)yy(f )xx(gyyxx1111

++++++ = 0 (♠)

Proof) Let the chord meet the circle at points A )y,x( 22 and B )y,x( 33

Equation of tangent

c)yy(f )xx(gyyxx 2222 ++++++ = 0 (∗)

P lies on (∗),

i.e. c)yy(f )xx(gyyxx 21212121 ++++++ = 0

It follows that )y,x( 22 satisfies equation (♠).

i.e. (♠) pass through the point A. Similarly, point B also satisfies (♠)

∴ (♠) is equation AB.

8. Length of PA = cfy2gx2yx11

2

1

2

1++−+

Proof) PA = 2

21

2

21 )yy()xx( −+−

= 2

221

2

1

2

221

2

1yyy2yxxx2x +−++−

Sub 2x in cfy2gx2yx 22 ++++ = 0 and 1x in (∗)

PA = cfy2gx2yx11

2

1

2

1++−+


slope: m

slope: m

A (x2, y2)

P (x1, y1) B (x3, y3)




9. Given the equations of 2 circles 1C and ,C2

the radical axis is .CC 21 −

10. Of the line Ax + By + c = 0 cuts the circle cfy2gx2yx 22 ++++ = 0

at 2 points P and Q, then any circle passing through P and Q has the form

)cByAx(k cfy2gx2yx22 +++++++ = 0.

11. Of the two circles intersect at P and Q, any circle passing through P and Q as

the form 21kCC + = 0.

12. Parametric form of 22 yx + = 2r

θ=θ=

sinr y

cosr x

Example 15 Let )y,x(P),y,x(P 222111 be two distinct points on the circle 22 yx + = 2r

2C

(a) Show that the equation of chord 21PP is)yy)(yy()xx)(xx(2121

−−+−− = 222 r yx −+

(b) Deduce that the equation of tangent at 1P is 11

yyxx + = 2r


radical axis




Example 16 Consider the line L: y = 2a and the circle C: 22 yx + = ,a 2 a > 0. Let P be a

variable point on L. If the tangents for P to C touch the circle C at points Q

and R respectively, show that the mid-point of QR lies on a fixed circle andfind centre and radius of this circle.

Example 17 Let P be the point outside the circle 22 yx + = 2r and A and B are points on

the circle such that PAB is straight line. Let Q be a point on the line PAB

such that PA : AQ = k : 1, set P ),y,x( 11 show2

1

2

1

k 1

kyy

k 1

kxx

++

+

++

= 2r

Hence, show the equation of tangents from the point P to the circle is

)r yx)(r yx( 22222

1

2

1−+−+ =

22

11)r yyxx( −+





Example 18 (a) Show for any real value m, the straight lines y = 2Mmamx ± are

tangents to the circle 22 yx + = .a2

(b) P is a variable point outside the circle 22 yx + = .a2 If two tangents are

drawn from P to the circle are ⊥, show the locus of P is also a circle.





9.6 ParabolaStandard form: 2y = 4ax

Parametric equation: x = 2at

y = 2atExample 19 The point t is one of the extremities of a focal chord of the parabola

2y = 4ax, prove the other extremity is the point ,t

1− and hence show the

locus of mid-point of the focal chord is a parabola.

Example 20 Prove that the equation of chord of the parabola 2y = 4ax with end points

)y,x( 11 and )y,x( 22 is )yy(y 21 − = .yyax4 21+


F (a, 0)

y2 = 4ax

focal chord

latus rectum

chord

x = −a

directrix

x

y




Remark:

If 21yy + = 0,x = 1x = 2x

Thm. Equation of tangent at )y,x( 11 for the parabola 2y = 4ax is

yy1 = )xx(a21

+

In parametric form: )at2,at( 2

2aty = )atx(a2 2+

y = att

x+

Example 21 Show that two tangents to a parabola 2y = 4ax are ⊥ each other iff the

intersection of the two tangents lies on the directrix (x = −a).





Example 22 Let 1P be parabola 2y = 4ax. A straight line L cuts the 1

P at 2 points A and

B. If M (α, β) is the mid-point of AB,

(a) find in terms of a, α, β, equation of L.

(b) Hence, find the locus of mid-points of chord 1P that are tangent to

another parabola2y:Pα = −4ax.





9.7 Ellipse

Standard form:2

2

2

2

b

y

a

x+ = 1

e: eccentricity

2e =2

2

a

b1− for a > b

If centre is (h, k), equation:2

2

2

2

b

)k y(

a

)hx( −+

−= 1

Parametric form: x = a cos θ

y = b sin θ

Example 23 Let P )y,x( 11 and Q )y,x( 22 be two points on2

2

2

2

b

y

a

x+ = 1. Prove

equation of PQ is


(ae, 0)(−ae, 0)

y

x

x = a/ex = −a/ea

0

b

chord

directrix

focal chord

latus rectum




)yy( b

y)xx(

a

x212212

+++ = 1 b

yy

a

xx2

21

2

21 ++

Remark:

As ,xx 12 → equation of tangent

)y2( b

y)x2(

a

x1212

+ = 1 b

y

a

x2

2

1

2

2

1 ++

= 2

2

1

2

1

b

yy

a

xx+ = 1

In parametric form: θ+θ sin b

ycos

a

x= 1

Example 24 P, Q are the points with parameter θ, φ on ellipse2

2

2

2

b

y

a

x + = 1. TP, TQ are

tangents to the ellipse. Find the coordinate T and show that if P and Q

moves as θ − φ = .2

π

Find the locus of T.

Example 25 If lines 2x − y + 3 = 0 and 2x + 3y + 7 = 0 both touch the ellipse

22 byax + = 1, find a, b.





Example 26 Two perpendicular tangents are drawn from an external point P to the

ellipse2

2

2

2

b

y

a

x+ = 1. Find locus of P.





9.8 Hyperbola

Standard form:2

2

2

2

b

y

a

x− = 1

2e =2

2

a

b1+ > 1

Asymptotes: b

y

a

x ± = 0

If centre is (h, k), equation:2

2

2

2

b

)k y(

a

)hx( −−

−= 1

Parametric form: x = a sec θ

y = b tan θ

Example 27 Prove the equation of chord of hyperbola2

2

2

2

b

y

a

x− = 1 at point

)y,x(),y,x( 2211 is )yy(

b

y)xx(

a

x212212

+−+ = .1

b

yy

a

xx2

21

2

21 +−


F(ae, 0)

F′

(−ae, 0)

0

y asymptote

x




9.9 General Equation of Second Degreecfy2gx2 byhxy2ax 22 +++++ = 0

Tangent at )y,x( 11

c)yy(f 2)xx(g2y by)yxxy(h2xax 11

2

111

2

1 ++++++++ = 0

If 2hab− = 0parabola

If 2hab− > 0 ellipse

If 2

hab− < 0 hyperbola

Thm. Let 2hab − ≠ 0, the quadratic curve cfy2gx2 byhxy2ax 22 +++++ = 0 will be

translated to a point (h, k) such that h = ,abh

hf bg2 −

−

k =abh

hgaf 2 −−

and the equation becomes cy byxh2xa 22′+′+′′+′ = 0.

Thm. Rotate the axis with angle θ (Reduce xy-term)

i.e. cfy2gx2 byhxy2ax 22 +++++ = 0

⇒ cyf 2xg2y bxa 22 ′+′+′+′+′ = 0 such that

1. tan 2θ = , ba

h2

− a − b ≠ 0

2. a − b = 0, θ =4

Example 27 7y2x10yxy4x 22 +−++− = 0 reduce the above equation in standard form by

suitable transformation of coordinates axes. Sketch it.

Prepared by K F Ngai

Two Dimensional Co-Ordinate Geometry

Documents

Transcript of Two Dimensional Co-Ordinate Geometry