FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

Class 9 Coordinate Geometry - Basics, Problems & Solved Examples | Math Square

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

CONIC SECTIONS

Definition

Conic sections or conics are the sections whose ratios of the distance of the variables point from the fixed point to the distance, or the variable point from the fixed line is constant

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

word image 3569

TYPES OF CONIC SECTION

There are     i) Parabola

ii) Ellipse

                   iii) Hyperbola

IMPORTANT TERMS USED IN CONIC SECTION

I.        FOCUS

This is the fixed point of the conic section.

For parabola

         

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

S → focus

          For ellipse

         

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

                                       S and S’ are the foci of an ellipse

II.  DIRECTRIX

This is the straight line whose distance from the focus is fixed.

For parabola

         

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

 

For ellipse

         

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

III.      ECCENTRICITY (e)

This is the amount ratio of the distance of the variable point from the focus to the distance of the variables point from the directrix.

For Parabola

 

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

 

For ellipse

         

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

 

word image 3570

word image 3571

 

word image 1265

 

 

IV.   AXIS OF THE CONIC

This is the straight line which cuts the conic or conic section symmetrically into two equal parts.

For parabola

 

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

 

X-axis is the point of the conic i.e. parabola

Also

 

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

Y-axis is the axis of the conic

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

FOR ELLIPSE

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

          AB – is the axis (major axis) of the Conic i.e. (ellipse)

CD – is the axis (minor axis) the Conic i.e. (ellipse)

→An ellipse has Two axes is major and minor axes

V    FOCAL CHORD

This is the chord passing through the focus of the conic section.

For parabola

 

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

 

For ellipse

 

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

VI  LATUS RECTRUM

This is the focal cord which is perpendicular to the axis of the conic section.

For parabola

 

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

 

For Ellipse

 

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

 

          Note:

          Latus rectum is always parallel to the directrix

VII.  VERTEX

This is the turning point of the conic section.

For parabola

 

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

0 – is the vertex

For ellipse

 

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

Where V and V1 is the vertex of an ellipse

PARABOLA

This is the conic section whose eccentricity, e is one i.e. e = 1

 

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

word image 3572

                    For parabola

word image 3573

SP = MP

      EQUATIONS OF THE PARABOLA

These are;

a) Standard equation

b) General equations

A. STANDARD EQUATION OF THE PARABOLA

1st case: Along the x – axis

·         Consider the parabola whose focus is S(a, 0) and directrix x = -a

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

word image 3574

          Squaring both sides

 

word image 3575

word image 1278
word image 3576

 

word image 3577

word image 3578
word image 3579
word image 3580

Is the standard equation of the parabola

PROPERTIES

i) The parabola lies along x – axis

ii) Focus, s (a, o)

iii) Directrix x = -a

iv)  Vertex (0, 0) origin

Note:

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

PROPERTIES

1) The parabola lies along x – axis

2) Focus s (-a, o)

3) Directrix x = a

4) Vertex (o, o) origin

2nd case: along y – axis

Consider the parabola when focus is s (o, a) and directrix y = -a

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

 

word image 3581

word image 3582
word image 3583
word image 3584

 

word image 3585

word image 3586
word image 3587
word image 3588

·         Is the standard equation of the parabola along y – axis

PROPERTIES

i) The parabola lies along y – axis

ii) The focus s (o, a)

iii)  Directrix y = -a

iv)  Vertex (o, o) origin

Note;

 

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

 

Hence,    x2 = -4ay

PROPERTIES

i) The parabola lies along y – axis

ii) Focus s (o, -a)

iii) Directrix y = a

iv) Vertex (o, o)

GENERAL EQUATION OF THE PARABOLA

·         Consider the parabola whose focus is s (u, v) and directrix ax + by + c = 0

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

 

 

word image 3589

word image 3590
word image 3591
word image 3592

          Is the general equation of the parabola

Where;

S (u, v) – is the focus

Examples:

1.  Find the focus and directrix of the parabola y2 = 8x

Solution

Given y2 = 8x

Comparing from

 

word image 3593

word image 3594
word image 3595
word image 3596
word image 3597
word image 3598
word image 3599
word image 3600

 

2.  Find the focus and the directrix of the parabola

y2 = -2x

Solution

word image 3601

Compare with

 

word image 3605

word image 3608
word image 3609

3.  Find the focus and directrix of x2 = 4y

Solution

 

word image 3610

word image 3611
word image 3612
word image 3613
word image 3614
word image 3615
word image 3616
word image 3617
word image 3618

4. Given the parabola x2 =

word image 3619

a) Find i) focus

ii) Directrix

iii) Vertex

b) Sketch the curve

Solution

 

word image 3620

word image 3621
word image 3622
word image 3623
word image 3624

i) Focus =

word image 3625

word image 3626

ii) Directrix, y = a

 

word image 3627

iii) Vertex

word image 3628

 

b) Curve sketching

         

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

 

5.  Find the equation of the parabola whose focus is (3, 0) and directrix

X = -3

Solution

Given focus (3, 0)

Directrix, x = -3

(3, 0) = (a, 0)

From

y2 = 4 (3)x

word image 3629

word image 3630

6.   Find the equation of the parabola whose directrix, y =

word image 3631

Solution

Given directrix, y =

word image 3632

Comparing with

 

word image 3633

word image 3634
word image 3635
word image 3636
word image 3637
word image 3638

 

7.  Find the equation of the parabola whose focus is (2, 0) and directrix, y = – 2.

Solution

Focus = (2, 0)

Directrix y = -2

 

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

 

word image 3639

word image 3640
word image 3641
word image 3642
word image 3643

 

8.   Find the equation of the parabola whose focus is (-1, 1) and directrix x = y

Solution

Given: focus = (-1, 1)

Directrix x = y

 

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

word image 3644
word image 3645

 

word image 3646

word image 3647

PARAMETRIC EQUATION OF THE PARABOLA
The parametric equation of the parabola  are given

word image 3648

word image 3649
word image 3650
word image 3651

X = at2 and y = 2at

Where;

t – is  a parameter

TANGENT TO THE PARABOLA

Tangent to the parabola, is the straight line which touches it at only one point.

 

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

Where, p – is the point of tangent or contact

CONDITIONS FOR TANGENT TO THE PARABOLA

a) Consider a line y = mx + c is the tangent to the parabola y2 = 4ax2. Hence the condition for tangency is obtained is as follows;

i.e.

 

word image 3652

word image 3653
word image 3654
word image 3655
word image 3656
word image 3657
word image 3658
word image 3659

 

word image 3660

word image 3661
word image 3662
word image 3663
word image 3664
word image 3665
word image 3666
word image 3667
word image 3668

 

word image 3669

word image 3670
word image 3671
word image 3672

b) Consider the line ax + by + c = is a tangent to the parabola y2 = 4ax Hence, the condition for tangency is obtained as follows;

i.e.

 

word image 3673

word image 3674
word image 3675
word image 3676
word image 3677
word image 3678
word image 3679
word image 3680
word image 3681
word image 3682
word image 3683
word image 3684
word image 3685

Examples

1.  Prove that the parametric equation of the parabola are given by

X = at2, and y = 2at

Solution

Consider the line

Y = mx + c is a tangent to the parabola y2 = 4ax. Hence the condition for tangency is given by y2 = 4ax

 

word image 3686

word image 3687
word image 3688
word image 3689
word image 3690
word image 3691
word image 3692
word image 3693
word image 3694
word image 3695
word image 3696

 

word image 3697

word image 3698
word image 3699
word image 3700

 

word image 3701

word image 3702
word image 3703

 

word image 3704

word image 3705
word image 3706
word image 3707
word image 3708
word image 3709
word image 3710

The parametric equation of the parabola of m is given as x = at2 and y = 2at

Where;

t – is a parameter

GRADIENT OF TANGENT OF THE PARABOLA

The gradient of tangent to the parabola can be expressed into;

i) Cartesian form

ii) Parametric form

i) IN CARTESIAN FORM

– Consider the tangent to the parabola y2 = 4ax Hence, from the theory.

Gradient of the curve at any = gradient of tangent to the curve at the point

 

word image 3711

word image 3712
word image 3713
word image 3714
word image 3715
word image 3716

ii) IN PARAMETRIC FORM

Consider the parametric equations of the parabola

i.e.

FORM SIX MATHEMATICS COORDINATE GEOMETRY II

word image 3717

word image 3718
word image 3719
word image 3720
word image 3721
word image 3722
word image 3723
word image 3724
word image 3725
word image 3726

EQUATION OF TANGENTS TO THE PARABOLA

These can be expressed into;

i) Cartesian form

ii) Parametric form

i) In Cartesian form

– Consider the tangent to the parabola y2 = 4ax at the point p (x, y)

 

FORM SIX MATHEMATICS STUDY NOTES COORDINATE GEOMETRY II

 

Hence the equation of tangent is given by

 

word image 3727

word image 3728
word image 3729
word image 3730
word image 3731
word image 3732
word image 3733
word image 3734
word image 3735
word image 3736

 

ii) In parametric form

·         Consider the tangent to the parabola y2 = 4ax at the point p (at2, 2at)

word image 1288

Hence the equation of tangent is given by;

 

word image 1289

word image 3737
word image 3738

 

word image 3739

word image 3740
word image 3741

Examples

1. Show that the equation of tangent to the parabola y2 = 4ax at the point

word image 3742

2. Find the equation of tangent to the parabola y2 = 4ax at (at2, 2at)

NORMAL TO THE PARABOLA

Normal to the parabola is the line which is perpendicular at the point of tangency.

word image 1290

Where;

P is the point of tangency

 

GRADIENT OF THE NORMAL TO THE PARABOLA

This can be expressed into;·

i) Cartesian form

ii) Parametric form

i) In Cartesian form

– Consider the gradient of tangency in Cartesian form

i.e.

word image 3743

Let M = be gradient of the normal in Cartesian form but normal is perpendicular to tangent.

 

word image 3744

word image 3745
word image 3746
word image 3747

ii) In Parametric form

Consider the gradient of tangent in parametric form.

word image 3748

Let m be gradient of the normal in parametric form.

But

Normal is perpendicular to the tangent

 

word image 3749

word image 3750
word image 3751
word image 3752

EQUATION OF THE NORMAL TO THE PARABOLA

These  can  be expressed into;·
i) Cartesian form

ii) Parametric form

 

i) In Cartesian form

Consider the normal to the point y2= 4ax at the point p (x1, y1) hence the equation of the normal given by;

 

word image 3753

word image 3754
word image 3755
word image 3756
word image 3757
word image 1291

 

ii) In parametric form

Consider the normal to the parabola y2 = 4ax at the point p (at2, 2at). Hence the equation of the normal is given by;

 

word image 3758

word image 3759
word image 3760
word image 3761
word image 3762

 

word image 3763

word image 3764

 

Examples:

1.  Find the equation of the normal to the parabola y2 =  at the point

word image 3765

word image 3766

2. Show that the equation of the normal to the parabola y2 = 4ax at the point (at3, 2at) is

word image 3767

 

CHORD TO THE PARABOLA

·         This is the line joining two points on the parabola

 

word image 1292

Let m – be gradient of the chord
Hence

ii) GRADIENT OF THE CHORD IN PARAMETRIC FORM
Consider a chord to the parabola  at the points  and

word image 3768

word image 3769
word image 3770
word image 3771
word image 3772
word image 1293
word image 3773
word image 3774
word image 3775
word image 3776
word image 3777
word image 3778
word image 3779
word image 3780
word image 3781
word image 1294

 

 

word image 3782

word image 3783
word image 3784
word image 3785

EQUATION OF THE CHORD TO THE PARABOLA.

These can be expressed into;·
i) Cartesian form
ii) Parametric form

i) EQUATION OF THE CHORD IN PARAMETRIC FORM

– Consider the chord to the parabola y2 = 4ax at the points. Hence the equation of the chord is given by;

word image 3786

word image 3787

 

word image 3788

word image 3789

 

word image 3790

word image 3791
word image 3792
word image 3793
word image 3794

II. EQUATION OF THE CHORD IN CARTESIAN FORM.

Consider the chord to the parabola y2 = 4ax at the point P1(x1, y1) and P2 (x2, y2) hence the equation of the chord is given by

word image 3795

word image 3796
word image 3797
word image 3798
word image 3799
word image 3800
word image 3801
word image 3802
word image 1295
word image 3803
word image 3804

EXCERSICE.

1.  Show that equation of the chord to the parabola y2 = 4ax at (x1, y1) and (x2, y2) is

word image 3805

2.  Find the equation of the chord joining the points () and

word image 3806

word image 3807

3.  As , the chord approaches the tangent at t1.deduce the equation of the tangent from the equation of the chord to the parabola y2 = 4ax.

word image 3808

THE LENGTH OF LATUS RECTUM

Consider the parabola

word image 3809

word image 1296
word image 3810
word image 3811

 

word image 3812

word image 3813

          Now consider another diagram below

word image 1297

Therefore, the length of latus rectum is given by

 

word image 3814

 

word image 3815

word image 3816
word image 3817

EQUATION OF LATUS RECTUM
– The extremities of latus rectum are the points p1 (a, 2a) and

p2 (a1, -2a) as shown below

 

word image 1298

Therefore, the equation of latus rectum is given by

OPTICAL PROPERTY OF THE PARABOLA
Any ray parallel to the axis of the parabola is reflected through the focus. This property which is of considerable practical use in optics can be proved by showing that the normal line at the point ‘’p’’ on the parabola bisects the angle between  and the line  which is parallel to the axis of the parabola.
Angle of INCIDENCE and angle of REFLECTION are equal

word image 3818

word image 3819
word image 3820
word image 3821
word image 3822
word image 3823
word image 3824
word image 3825
word image 3826

 

word image 1299

 

– is the normal line at the point ‘p’ on the parabola
i.e.

Note that;  (QPS) Is an angle.

Examples
Prove that rays of height parallel to the axis of the parabolic mirror are reflected through the focus.

word image 3827

word image 3828
word image 3829
word image 3830
word image 3831
word image 3832
word image 3833
word image 3834
word image 3835
word image 3836
word image 3837
word image 3838
word image 3839
word image 3840
word image 3841
word image 3842
word image 3843
word image 3844
word image 3845
word image 3846
word image 3847
word image 3848
word image 3849
word image 3850
word image 3851
word image 3852
word image 3853

TRANSLATED PARABOLA

1.

word image 3854

– consider the parabola below

 

word image 1300

PROPERTIES.

I) The parabola is symmetrical about the line y = d through the focus
II) Focus,
III) Vertex,
IV) Directrix,

word image 3855

word image 3856
word image 3857

2.

word image 3858

– Consider the parabola below

word image 1301

 

PROPERTIES

I) the parabola is symmetrical about the line x = c, through the focus
II) Focus
III) Vertex,
IV) Directrix,

word image 3859

word image 3860
word image 3861

Examples

1. Show that the equation  represent the parabola and hence     find

word image 3862

i) Focus

ii) Vertex

iii) Directrix

iv) Length of latus rectum

Solution

Given;

 

word image 3863

word image 3864
word image 3865
word image 3866

 

word image 3867

word image 3870

 

word image 3871

word image 3872
word image 3873
word image 3874

 

word image 3875

word image 3876
word image 3877
word image 3878
word image 3879
word image 3880
word image 3881
word image 3882
word image 3883
word image 3884
word image 3885
word image 3886
word image 3887
word image 3888

FORM SIX MATHEMATICS COORDINATE GEOMETRY II

word image 1302

word image 3889
word image 3890
word image 3891
word image 3892
word image 3893
word image 3894
word image 3895

2. Shown that the equation x2 + 4x + 2 = y represents the parabola hence find its focus.

Solution

Given;

 

word image 3896

word image 3897
word image 3898
word image 3899

 

word image 3900

word image 3901
word image 3902
word image 3903
word image 3904
word image 3905

 

word image 3906

word image 3907
word image 3908
word image 3909

 

word image 3910

word image 3911
word image 3912

3.  Show that the equation x2 + 4x – 8y – 4 = 0 represents the parabola whose focus is at (-2, 1)

Solution

 

word image 3913

word image 3914
word image 3915
word image 3916
word image 3917
word image 3918
word image 3919
word image 3920
word image 3921

 

word image 1303

word image 3922
word image 3923

 

word image 3924

word image 3925
word image 3926
word image 3927
word image 3928
word image 3929
word image 3930
word image 3931

ELLIPSE

This is the conic section whose eccentricity e is less than one

I.e. |e| < 1

         

word image 1304

 

word image 3932

word image 3933
word image 3934

AXES OF AN ELLIPSE

An ellipse has two axes these are
i) Major axis
ii) Minor axis

1.  MAJOR AXIS

Is the one whose length is large

2.   MINOR AXIS

Is the one whose length is small

a)

word image 1305

 

b)

word image 1306

 

Where

AB – Major axis

PQ – Minor axis

EQUATION OF AN ELLIPSE
These are;
i) Standard equation

ii) General equation

1.   STANDARD EQUATION

– Consider an ellipse below;

 

word image 1307

word image 3935

 

word image 3936

word image 3937
word image 3938
word image 3939
word image 3940
word image 3941
word image 3942
word image 3943
word image 3944
word image 3945

 

word image 3946

word image 3947
word image 3948
word image 3949
word image 3950
word image 3951
word image 3952
word image 3953
word image 3954
word image 3955
word image 3956
word image 3957

1st CASE
Consider an ellipse along x – axis

word image 1308

 

 

word image 3958

word image 3959
word image 3960
word image 3961
word image 3962
word image 3963
word image 3964
word image 3965

FORM SIX MATHEMATICS COORDINATE GEOMETRY II

word image 3966

word image 3967
word image 3968
word image 3969
word image 3970
word image 3971
word image 3972
word image 3973
word image 3974
word image 3975

PROPERTIES

I) an ellipse lies along the x – axis (major axis)

ii) a > b

iii)

word image 3976

iv) Foci,

word image 3977

v)  Directrix

word image 3978

vi) Vertices, (a, o), (-a, o) along major axis

(0, b) (0, -b) along minor axis

vi) The length of the major axis l major = 2a

viii) Length of minor axis l minor = 2b

Note:

For an ellipse (a – b) the length along x – axis

B – is the length along y – axis

2nd CASE

·         Consider an ellipse along y – axis

word image 1309

 

word image 3979

word image 3980
word image 3981
word image 3982
word image 3983
word image 3984
word image 3985

 

word image 3986

word image 3987
word image 3988
word image 3989
word image 3990
word image 3991

 

word image 3992

word image 3993
word image 3994
word image 3995

        

word image 3996

PROPERTIES

i) An ellipse lies along y – axis (major axis)

ii) b > a

iii)

word image 3997

iv) Foci

word image 3998

v) Directrices

word image 3999

vi) Vertices:  = along major

word image 4000

= along minor as

word image 4001

vii) Length of the major axis L major = 2b

viii) Length of the minor axis   L minor = 2a

II. GENERAL EQUATION OF AN ELLIPSE

·         Consider an ellipse below y – axis

word image 1310

From

EXAMPLE
Given the equation of an ellipse
Find i) eccentricity
ii) Focus
iii) Directrices

word image 4002

word image 4003
word image 4004
word image 4005
word image 4006
word image 4007
word image 4008
word image 4009

Solution

Given

word image 4010

Compare from

 

word image 4011

word image 4012
word image 4013
word image 4014
word image 4015
word image 4016
word image 4017
word image 4018
word image 4019

 

word image 4020

word image 4021

 

word image 4022

word image 4023
word image 4024
word image 4025
word image 4026
word image 4027
word image 4028
word image 4029

Find the focus and directrix of an ellipse 9x2 + 4y2 = 36

Solution

Given;

 

word image 4030

word image 4031
word image 4032
word image 4033
word image 4034
word image 4035
word image 4036
word image 4037
word image 4038

 

word image 4039

word image 4040
word image 4041
word image 4042
word image 4043
word image 4046
word image 4047
word image 4048
word image 4049

CENTRE OF AN ELLIPSE
This is the point of intersection between major and minor axes

word image 1311

·         O – Is the centre of an ellipse

word image 1312

A – Is the centre of an ellipse

DIAMETER OF AN ELLIPSE.

This is any chord passing through the centre of an ellipse

word image 1313

Hence  – diameter (major)
– Diameter (minor)
Note:
i) The equation of an ellipse is in the form of

ii) The equation of the parabola is in the form of

iii) The equation of the circle is in the form of
word image 4050

word image 4051
word image 4052
word image 4053
word image 4054
word image 4055

PARAMETRIC EQUATIONS OF AN ELLIPSE

The parametric equations of an ellipse are given as

And

word image 4056

word image 4057

Where

θ – Is an eccentric angle

Recall

 

word image 4058

word image 4059
word image 4060
word image 4061
word image 4062
word image 4063
word image 4064

TANGENT TO AN ELLIPSE
This is the straight line which touches the ellipse at only one point

word image 1314

Where;
P – Is the point of tangent or contact
Condition for tangency to an ellipse
Consider the line b = mx + c is the tangent to an ellipse

word image 4065

word image 4066
word image 1315
word image 4067
word image 4068
word image 4069
word image 4070
word image 4071
word image 4072
word image 4073
word image 4074
word image 4075
word image 4076
word image 4077
word image 4078
word image 4079
word image 4080

Examples
Show that, for a line  to touch the ellipse        Then,

word image 4081

word image 4082

GRADIENT OF TANGENT TO AN ELLIPSE
This can be expressed into;
i) Cartesian form
ii) Parametric form

word image 4083

1. GRADIENT OF TANGENT IN CARTESIAN FORM

– Consider an ellipse

word image 4084

Differentiate both sides with w.r.t x

 

word image 4085

word image 4086
word image 4087
word image 4088
word image 4089

ii. GRADIENT OF TANGENT IN PARAMETRIC FORM

– Consider the parametric equation of an ellipse

 

word image 4090

word image 4091
word image 4092
word image 4093
word image 4094

 

word image 4095

word image 4096
word image 4097
word image 4098
word image 4099

EQUATION OF TANGENT TO AN ELLIPSE

These can be expressed into;
i) Cartesian form
ii) Parametric form

I.   Equation of tangent in Cartesian form

– Consider the tangent an ellipse

word image 4100

word image 1316

                    Hence, the equation of tangent is given by

 

word image 4101

word image 4102
word image 4103
word image 1317

 

word image 4104

word image 4105
word image 4106

 

word image 4107

word image 4108
word image 4109

 

word image 4110

word image 4111
word image 4112

EQUATION OF TANGENT IN PARAMETRIC FORM.

Consider the tangent to an ellipse  At the point

word image 4113

word image 4114
word image 1318

Hence the equation of tangent is given by

 

word image 4115

word image 4116
word image 4117
word image 4118

 

word image 4119

word image 4120
word image 4121
word image 4122
word image 4123
word image 4124

 

word image 4125

word image 4126

Note
1.

2.

word image 4127

word image 4128
word image 4129
word image 1319
word image 4132
word image 4133
word image 4134

EXERCISE
i. Show that the equation of tangent to an ellipse
ii.  Show that the equation of tangent to an ellipse
iii.  Show that the gradient of tangent to an ellipse
NORMAL TO AN ELLIPSE
Normal to an ellipse perpendicular to the tangent at the point of tangency.

word image 4135

word image 4136
word image 4137
word image 1320

                Where: p is the point of tangency

GRADIENT OF THE NORMAL TO AN ELLIPSE.

This can be expressed into two
i) Cartesian form
ii) Parametric

I) IN CARTESIAN FORM

– Consider the gradient of the tangent in Cartesian form

But normal tangent

 

word image 4138

word image 4139

 

word image 4140

word image 4141
word image 4142
word image 4143
word image 4144

II) IN PARAMETRIC FORM
Consider the gradient of tangent in parametric form

Let m = slope of the normal in parametric form

EQUATION OF THE NORMAL TO AN ELLIPSE
This can be expressed into;
(i) Cartesian form
(ii)  Parametric form

word image 4145

word image 4146
word image 4147
word image 4148

I. IN CARTESIAN FORM

– consider the normal to an ellipse

word image 4149

 

word image 1321

 

Hence the equation of the normal is given by

word image 4150

FORM SIX MATHEMATICS COORDINATE GEOMETRY II

word image 4151

word image 4152
word image 4153
word image 4154

II) IN PARAMETRIC FORM
Consider the normal to an ellipse

word image 4155

word image 1322

Hence the equation of the normal is given by

Examples
·         Show that the equation of the normal to an ellipse
word image 4156

word image 4157
word image 4158
word image 4159
word image 4160
word image 4161
word image 4162
word image 4163
word image 4164
word image 4165

CHORD OF AN ELLIPSE.

This is the line joining any two points on the curve ie (ellipse)

word image 1323

GRADIENT OF THE CHORD TO AN ELLIPSE.

This can be expressed into
i) Cartesian form
ii) Parametric form

I.  IN CARTESIAN FORM

– Consider the point A (x1, y1) and B (x2, y2) on the ellipse  hence the gradient of the cord is given by

II. IN PARAMETRIC FORM

Consider the points A  and B  on the ellipse  Hence the gradient of the chord is given by;

word image 4168

word image 4169
word image 4170
word image 4171
word image 4172
word image 4173
word image 4174

 

EQUATION OF THE CHORD TO AN ELLIPSE

These can be expressed into
i) Cartesian form
ii) Parametric form
I:  IN CARTESIAN FORM.

Consider the chord the ellipse at the point A (x1, y1) and B(x2,y2). Hence the equation of the chord is given by;

word image 4175

word image 4176
word image 4177
word image 4178
word image 1324
word image 4179
word image 4180

II.   IN PARAMETRIC FORM.

Consider the chord to an ellipse  at the points. Hence the equation of the chord is given by

word image 4181

word image 4182
word image 4183
word image 4184
word image 4185
word image 4186
word image 4187
word image 4188
word image 4191
word image 4192

FOCAL CHORD OF AN ELLIPSE.

This is the chord passing through the focus of an ellipse

word image 1325

Where  = is the focal chord
Consider the points A and B are respectively  Hence
Gradient of AS = gradient of BS
Where s = (ae, o)

word image 4193

word image 4194
word image 4195
word image 4196
word image 4200
word image 4201
word image 4202
word image 4203
word image 4204
word image 4205
word image 4206

DISTANCE BETWEEN TWO FOCI.

Consider the ellipse below;

word image 1326

2

word image 4207

2

word image 4208

2

word image 4209

 

                                               

word image 4210

Where a = is the semi major axis

e = is the eccentricity

DISTANCE BETWEEN DIRECTRICES.

Consider the ellipse below

           

word image 1327

 

 

word image 4211

=

word image 4212

=

word image 4213

 

word image 4214

Where a – is the semi major axis
e – is the eccentricity

 

LENGTH OF LATUS RECTUM.

Consider the ellipse below

word image 1328

word image 4215

 

word image 4216

word image 4217
word image 4220
word image 4221
word image 4222
word image 4223

 

word image 4224

word image 4225
word image 4226

 

word image 4227

word image 4228
word image 4229
word image 4230
word image 4231

 

word image 4232

word image 4233

IMPORTANT RELATION OF AN ELLIPSE

Consider an ellipse  below

 

word image 1329

 

word image 4234

word image 4235
word image 4236
word image 4237
word image 4238
word image 4239
word image 4240
word image 4241

 

word image 4242

word image 4243
word image 4244

ECCENTRIC ANGLE OF ELLIPSE

.This is the angle introduced in the parametric equation of an ellipse
I.e
Where      – is an eccentric angle
CIRCLES OF AN ELLIPSE
These are 1) Director Circle
2) Auxiliary circle

1.   DIRECTOR CIRCLE
– This is the locus of the points of intersection of the perpendicular tangents.

word image 4245

word image 4246

         

word image 1330

Consider the line  is tangent to the ellipse
Hence

2.  AUXILIARY CIRCLE
– This is the circle whose radius is equal to semi – major axis

word image 4247

word image 4248
word image 4249
word image 4250
word image 4251
word image 4252
word image 4253
word image 4254
word image 4258
word image 4261
word image 4263
word image 4264
word image 4265
word image 4266
word image 4267
word image 4268

 

word image 1331

 

Using Pythagoras theorem

word image 4269

   a – is the radius of the auxiliary circle

FORM SIX MATHEMATICS COORDINATE GEOMETRY II

CONCENTRIC ELLIPSE.

These are ellipse whose centre are the same.

word image 1332

The equations of centric ellipse are;

word image 4270

Where  a and b semi – major and semi – minor axes of the small ellipse

A and B are the semi – major and semi – minor axes of the large ellipse

A – a = B – b

A – B = a – b·
Is the condition for concentric ellipse

 

TRANSLATED ELLIPSE

This is given by the equation

word image 4271

A. PROPERTIES

i) An ellipse lies along x – axis
ii) a > b
iii) Centre (h, k)
iv) Vertices
v) Eccentricity,
vi) Foci
vii) Directrices
B. PROPERTIES
i) An ellipse has along y – axis
ii) b > a
iii) Centre (h, k)
iv) Vertices
v) Eccentricity
vi) Foci

Examples
Show that the equation 4x2 – 16x + 9y2 + 18y – 11 = 0 represents an    ellipse and hence find i) centre ii) vertices iii) eccentricity iv) foci v) directrices.

word image 4272

word image 4273
word image 4274
word image 4275
word image 4276
word image 4277
word image 4278
word image 1333

Solution

 

word image 4279

word image 4280
word image 4281
word image 4282
word image 4283
word image 4284
word image 4285

 

word image 1334

word image 4287
word image 4288
word image 4289
word image 4290
word image 4291

 

word image 4292

word image 4293

 

word image 4294

word image 4295
word image 4296
word image 4297

FORM SIX MATHEMATICS COORDINATE GEOMETRY II

word image 4298

word image 4299
word image 4300
word image 4301
word image 4302
word image 4303

 

word image 4304

word image 4305
word image 4306
word image 4307
word image 4308
word image 4309

 

word image 4310

word image 4311
word image 1335
word image 4312
word image 4313

III.      HYPERBOLA

This is the conic section whose eccentricity ‘’e’’ is greater than one       ( e > 1)

word image 1336

The hyperbola has two foci and two directrices

 

word image 1337

Where S and S’ are the foci of the hyperbola hence

word image 4314

Where e > 1

EQUATION OF THE HYPERBOLA

There are;
i) Standard equation
ii) General equation

1.    STANDARD EQUATION OF THE HYPERBOLA

Consider

         

word image 1338

 

word image 4315

word image 4316
word image 4317
word image 4318
word image 4319
word image 4320

 

 

JE UNAMILIKI SHULE AU BIASHARA NA UNGEPENDA IWAFIKIE WALIO WENGI?BASI TUNAKUPA FURSA YA KUJITANGAZA NASI KWA BEI NAFUU KABISA BOFYA HAPA KUJUA

 

But for more post and free books from our site please make sure you subscribe to our site and if you need a copy of our notes as how it is in our site contact us any time we sell them in low cost in form of PDF or WORD.

 

 

UNAWEZA JIPATIA NOTES ZETU KWA KUCHANGIA KIASI KIDOGO KABISA:PIGA SIMU:0787237719

 

SHARE THIS POST WITH FRIEND

Leave a comment