# Ellipse

Analysis of the Ellipse; Its Tangents and the Auxiliary Circle Chords

**Contents**

- Definition
- Ellipse General Equation
- The Tangent To An Ellipse At A Given Point
- The Normal To The Tangent At (x',y')
- The Points Of Intersection Of A Straight Line And An Ellipse
- The Parametric Equation Of An Ellipse
- The Parametric Equation Of A Chord
- The Equation Of A Tangent
- The Meet Of Two Tangents At Points With Differing Eccentric Angles
- Page Comments

## Definition

An

An

**ellipse**is the locus of all points of the plane whose distances to two fixed points add to the same constant.An

**ellipse**can be obtained as a result from the intersection of a cone by a plane in a way that produces a closed curve.**Circles**are special cases of ellipses, obtained when the cutting plane is orthogonal to the cone's axis.## Ellipse General Equation

If*X*is the foot of the perpendicular from

*S*to the Directrix, the curve is symmetrical about the line

*XS*. This line is taken to be the

*x*axis.

- The
**ratio**,is called**eccentricity**and is less than 1 and so there are two points on the line*SX*which also lie on the curve. - One
*A'*will lie between between*S*and*X*and nearer*S*and the other*X*will lie on*XS*produced. - Let the distance
*AA'*be*2a*and let the mid point of*AA'*be the point*C*. *C*is called the Centre of the Ellipse and is taken to be the origin.

The

**eccentricity**of an**ellipse**is :
Since

*A*lies on the curve: Since*A'*lies on the curve: Adding But Therefore or Also We now the lengths of both*CX*and*CS*and can proceed to find the equation of the curve. from equation (2) we know thatThe

**area**of an**ellipse**is : This result can be proven by direct integration.
Therefore
and
also
Therefore
Therefore
or

Let and the equation of the Ellipse becomes:

The

**perimeter**of an**ellipse**can't be expressed in a simple form. One**approximation**for the**perimeter**is .
The length

**a**is called the**semi-major axis.**When is plus or minus**b**which is called the**semi-minor axis.**##### Notes

- The equation of an ellipse if the axis of length
*2a*is taken up the*y*axis is given by:

*2a*. i.e. The major axis is always the larger of the two.- The equation of an Ellipse referred to a parallel axes through the point is given by the equation:

- The equation of an ellipse if the axis of length

Example:

##### Example - Focal distances

Problem

Show that the sum of the focal distances of any point on an is Ellipse is equal to the length of the major axis. Deduce a simple mechanical method for constructing the curve

Workings

From the diagram and from the definition of an ellipse, if is the abscissa of
Similarly if is drawn perpendicularly to the second Directerix
The sum of the focal distances and is therefore equal to , the length of the major axis.

Solution

By fixing two pins at ans and keeping stretched, by a pencil point, an endless piece of string passing round the pins, the pencil will describe an ellipse with , as foci.

## The Tangent To An Ellipse At A Given Point

Differentiating the equation for an ellipse with respect to Hence the gradient at isThe equation of the tangent at is therefore given by: This can be written as:

Dividing through by and using the equation for an ellipse,which is the condition that the point shall lie on the ellipse, the equation of the tangent at the point can be written as:

## The Normal To The Tangent At (x',y')

The

**Normal**at the point is the line perpendicular to the tangent and therefore its slope is and its equation is:Example:

##### Example - Normal and Tangent

Problem

Find the equation of the Tangent and Normal to the Ellipse at the point in the first quadrant whose ordinate is .

Workings

The abscissae of points on the ellipse at which the ordinate is are given by putting in the equation of the ellipse. This gives i.e. . The point in the first quadrant with these coordinates is The equation of the ellipse can now be written in the form:

Solution

Thus and

The equation of the tangent is thus given by: The equation of the Normal is given by the equation

The equation of the tangent is thus given by: The equation of the Normal is given by the equation

## The Points Of Intersection Of A Straight Line And An Ellipse

The coordinates of the points of intersection of the straight line and the ellipse are the values of*x*and

*y*which simultaneously satisfy both equations. Writing in the equating of the ellipse: or This quadratic has real, equal or imaginary roots depending upon whether is positive,zero or negative. When , the line intersects the ellipse in two real roots. When , the line intersects the ellipse only in imaginary points and when the line is a tangent to the ellipse.

Writing in the equation of the line:
we get a line which always touches the ellipse. Also since the radical sign on the right hand side of the equation may be either positive or negative, there are two tangents parallel to each other.

Example:

##### Example - Locus of points

Problem

Find the locus of the points of intersection of tangents to an ellipse which are at right angles to each other.

Workings

Using the standard equation for a Ellipse:
And the equation for a tangent from above:
The equation of a perpendicular tangent at a gradient of is :
These two equations for perpendicular tangents to the ellipise can be written as:
and
The coordinates of the point of intersection of the tangents simultaneously satisfy these two equations. If m is eliminated from the two equations the locus of the pint of intersection will be found.
Squaring and adding:
or

Solution

The locus is therefore a circle with centre coincident with the centre of the ellipse and with a radius of . This circle is called

**the director circle**.## The Parametric Equation Of An Ellipse

It is often convenient to express the coordinates of any point on the ellipse in terms of one variable. Draw a circle on the major angle of an ellipse with a centre coinciding with the centre of the ellipse. This circle is called**The Auxiliary Circle**Draw a radius of this circle making an angle with the major axis. Then and the coordinates of are .Draw the perpendicular from to the major axis to meet the ellipse at . The -coordinate of is also and since the point lies on the ellipse: Therefore Thus

Therefore the Parametric equation of an ellipse is :

## The Parametric Equation Of A Chord

The gradient of the chord joining the points and is given by:**The**Equation of the Chord is therefore

## The Equation Of A Tangent

If the above equation for a Chord becomes the equation of a Tangent at the point .The equation is:

## The Meet Of Two Tangents At Points With Differing Eccentric Angles

Let the two points have eccentric angles of and . Therefore the equations of the tangents are:
and
Eliminating from these equations:
Therefore
Or
Therefore
Eliminating from the equations:
Therefore
Thus
But since the point of intersection of the Tangents is:

Example:

##### Example - Chord

Problem

The Chord of an Ellipse passes through the focus . Show that the meet of the tangents at and lies on the Directrix .

Workings

Suppose and

The equation of is : This passes through the point and so: The meet of the tangents at and is :

The equation of is : This passes through the point and so: The meet of the tangents at and is :

Solution

The coordinate is :