# Parabola

An analysis of the Parabola, its proprties, Chords and Tangents

**Contents**

- Definition
- To Find The Tangent Of Gradient M
- The Parametric Equation Of A Parabola.
- The Normal
- The Equation Of The Line Joining Two Points On A Parabola.
- The Parametric Equation Of The Tangent
- To Find The Equation Of The Tangent To The Parabola At A Given Point.
- The Locus Of The Foot Of The Perpendicular From The Focus To A Tangent.
- The Locus Of The Intersection Of Perpendicular Tangents
- The Polar Of (x',y')
- The Feet Of The Normals From A Point To The Parabola
- The Circle Through The Feet Of The Normals
- An Important Property Of The Parabola
- Page Comments

## Definition

**Parabola**is a

**conic**section, the intersection of a right circular conical surface and a plane parallel to a generating straight line of that surface .

**Conics**. These are defined by their focus-directrix property, i.e. as the locus of a point which moves so that its distance from a fixed point is in a constant ratio to its distance from a fixed line. The fixed point is called the

**FOCUS**, the fixed line the

**DIRECTERIX**and the constant ratio the

**ECCENTRICITY**.

- For the Parabola the eccentricity is 1.
- For the ellipse the eccentricity is less than 1.
- For the Hyperbola the eccentricity is greater than 1.

This is the equation of a

**PARABOLA**.## To Find The Tangent Of Gradient M

The

The

**tangent**line to a curve at a given point is the straight line that "just touches" the curve at that point.The

**gradient**of a scalar field is a vector field that points in the direction of the greatest rate of increase of the scalar field, and whose magnitude is the greatest rate of change.
Let the equation of the tangent be :
This will meet the Parabola where:
i.e. where
For a tangent , the roots of this equation must be coincident and therefore:

**The Tangent of Gradient m is**

Example:

##### Example - Example 1

Problem

Find the equations of the tangents and normals to the parabola at the points(16,16) and (1,-4). The tangents intersect at the point and the normals intersect at . Prove that the line is parallel to the axis of the Parabola.

Workings

Here so that . For the point (16,16) and the equation for the tangent and normal are :
and respectively.
These can be rearranged to give simpler forms
and
Similarly for the point (1- 4) the equations for the tangent and Normal are:
and
The coordinates of are found from the simultaneous equation based on the tangents
From which and
The coordinates of are similarly found from the two equations of the Normals and the solution is and .

Solution

Conclusion:

Thus both and are at a height 6 above the axis and the line is therefore parallel to the axis which is also the axis of the Parabola.

Thus both and are at a height 6 above the axis and the line is therefore parallel to the axis which is also the axis of the Parabola.

## The Parametric Equation Of A Parabola.

It is now possible to rewrite equation (1) using the value of c:- or
and by substitution

So the point is a parametric expression for a point on the Parabola for all values of and the gradient of the tangent at that point is . By writing for , it can be seen that the point also lies on the equation and the following table can be constructed.
NOTE. It is usual to identify a point on a Parabola by using the parametric form . However if the gradient of a tangent is required, then it is better to use . Whilst it is easier to find the gradient of the normal by expressing the point as .

Example:

##### Example - Focus example

Problem

A chord of a parabola passes through the focus. n Show that the tangents at and meet on the Directerix.

Workings

Let the coordinates of be and be
The equation of the chord is:
This line passes through the focus and putting and
or

Solution

The meet of the tangents at and is
The coordinate is or since and so the meet lies on the line which is the Directerix,.

## The Normal

**Normal**, to a flat surface is a vector that is perpendicular to that surface. A normal to a non-flat surface at a point on the surface is a vector perpendicular to the tangent plane to that surface at .

The Normal at may be written down from the knowledge that its gradient is . The equation of the normal will be in the form .
Substituting values for and and the gradient

Thus

**the****equation****of****the****Normal**is given by:## The Equation Of The Line Joining Two Points On A Parabola.

Let the two points be andThe equation is given by:
or

## The Parametric Equation Of The Tangent

As the point moves ever closer to on the parabola, the chord joining the points becomes closer to the tangent. The parametric equation of the tangent at point is found by putting equal to and is given by:or

## To Find The Equation Of The Tangent To The Parabola At A Given Point.

Let the point be and the equation of the Parabola beDifferentiating w.r.t x
It is therefore possible to write down the equation of the tangent that goes through as:
And since
then

## The Locus Of The Foot Of The Perpendicular From The Focus To A Tangent.

A

Two lines or planes (or a line and a plane) are considered

**Locus**is a collection of points which share a property.Two lines or planes (or a line and a plane) are considered

**perpendicular**to each other if they form congruent adjacent angles (a T-shape).
The equation of any tangent is given by:
If the Perpendicular to this tangent passes through , its equation is given by:-

We can find the

**locus**by eliminating from the two equations i e. From which it can be seen that is the tangent at the Vertex.## The Locus Of The Intersection Of Perpendicular Tangents

The equation of a tangent of gradient is: And the equation of the tangent who's gradient is is:The

**locus**is found by eliminating from the above equations: This is the equation of the Directerix.## The Polar Of (x',y')

Suppose that the point of contact of the tangents are and . Then the tangent at passes through
But this is equally the condition that the line should pass through . But since there is a unique line joining two points, must be the polar of

## The Feet Of The Normals From A Point To The Parabola

Suppose that the Normal at passes through the given point . Then satisfies the following equation:
This is a cubic in giving three values for . i.e. there are in general three Normals passing through a given point. Moreover since a cubic has either three or one real root, there must be three or one real Normal.
As the term in squared is missing, the sum of the roots is zero. Therefore the sum of the ordinates of the feet is zero and so the centre of gravity of the triangle formed by the feet of the normals from any point lies on the axis.

## The Circle Through The Feet Of The Normals

Consider the intersection of the Parabola with the general circle,
If they meet at the point
This is a quartic in and since the term in is missing the sum of the roots is zero. Therefore the sum of the ordinates of the points of intersection of the parabola with any circle is zero.
Since the sum of the ordinates of the feet of the Normals is already zero, the circle through the three feet of the Normals from any point to the parabola must pass through the origin.

## An Important Property Of The Parabola

In the diagram is the point t2 on the Parabola whose Focus is . is the tangent at meeting the axis at . Since the coordinates of and are respectively and The equation of the tangent is given by: This meets the -axis where: and since then Hence and the triangle is isosceles and the angle = angle If is drawn through P parallel to the axis, it follows that:
So that the lines and are equally inclined to the tangent at . If is the Normal at , it follows that the lines and are equally inclined to the normal at .
This is an important property of the parabola. It means that if a ray of light starting from the focus strikes a parabolic mirror at , the reflected ray, which makes an equal angle with the normal, will be parallel to the axis of the mirror. Since is any point on the parabola it follows that all incident rays from a source at the focus will be reflected as rays which are parallel to the mirrors axis. Conversely all parallel rays from a distant source striking a parabolic mirror will be reflected so as to pass through the focus.