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TOPIC 11: COORDINATE GEOMETRY II | MATHEMATICS FORM 6

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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

 

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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

 

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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

 

 

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          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

 

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4. Given the parabola x2 =

Word Image 3619

a) Find i) focus

ii) Directrix

iii) Vertex

b) Sketch the curve

Solution

 

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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.

 

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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.

 

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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

 

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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

 

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ii) IN PARAMETRIC FORM

Consider the parametric equations of the parabola

i.e.

FORM SIX MATHEMATICS COORDINATE GEOMETRY II

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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

 

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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

 

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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 9x+ 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

 

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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

 

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Word Image 4104 Word Image 4105 Word Image 4106

 

Word Image 4107 Word Image 4108 Word Image 4109

 

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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

 

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Word Image 4125 Word Image 4126

Note
1.

2.

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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.

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                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

 

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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

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I. IN CARTESIAN FORM

– consider the normal to an ellipse

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Hence the equation of the normal is given by

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FORM SIX MATHEMATICS COORDINATE GEOMETRY II

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II) IN PARAMETRIC FORM
Consider the normal to an ellipse

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Hence the equation of the normal is given by

Examples
·         Show that the equation of the normal to an ellipse
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CHORD OF AN ELLIPSE.

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

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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;

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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;

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II.   IN PARAMETRIC FORM.

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

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FOCAL CHORD OF AN ELLIPSE.

This is the chord passing through the focus of an ellipse

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

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DISTANCE BETWEEN TWO FOCI.

Consider the ellipse below;

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2

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2

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2

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Where a = is the semi major axis

e = is the eccentricity

DISTANCE BETWEEN DIRECTRICES.

Consider the ellipse below

           

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=

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=

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

 

LENGTH OF LATUS RECTUM.

Consider the ellipse below

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IMPORTANT RELATION OF AN ELLIPSE

Consider an ellipse  below

 

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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.

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Consider the line  is tangent to the ellipse
Hence

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

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Using Pythagoras theorem

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   a – is the radius of the auxiliary circle

FORM SIX MATHEMATICS COORDINATE GEOMETRY II

CONCENTRIC ELLIPSE.

These are ellipse whose centre are the same.

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The equations of centric ellipse are;

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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

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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.

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Solution

 

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FORM SIX MATHEMATICS COORDINATE GEOMETRY II

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III.      HYPERBOLA

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

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The hyperbola has two foci and two directrices

 

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

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Where e > 1

EQUATION OF THE HYPERBOLA

There are;
i) Standard equation
ii) General equation

1.    STANDARD EQUATION OF THE HYPERBOLA

Consider

         

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