ADVANCED MATHEMATICS FORM 6 – COORDINATE GEOMETRY II

PROPERTIES

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

ii) b > a

iii)

iv) Foci

v) Directrices 

vi) Vertices:  = along major

 = along minor as

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

From







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

Solution

Given

Compare from

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

Solution

Given;

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

·         O – Is the centre of an ellipse

A – Is the centre of an ellipse

DIAMETER OF AN ELLIPSE.

This is any chord passing through the centre of an ellipse

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

PARAMETRIC EQUATIONS OF AN ELLIPSE

The parametric equations of an ellipse are given as

    And  

Where

          θ – Is an eccentric angle

Recall

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

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

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

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

1. GRADIENT OF TANGENT IN CARTESIAN FORM

– Consider an ellipse

Differentiate both sides with w.r.t x

ii. GRADIENT OF TANGENT IN PARAMETRIC FORM

– Consider the parametric equation of an ellipse

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

                    Hence, the equation of tangent is given by

EQUATION OF TANGENT IN PARAMETRIC FORM.

Consider the tangent to an ellipse  At the point

Hence the equation of tangent is given by

Note
1.   

2.      

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.

                Where: p is the point of tangency