Equations can also be written in **parametric form**. In this form, *x* and *y* are both written in terms of a third variable called a parameter.

An example of a parametric equation is .

Try to write in a parametric form.

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Given parameter *p*:

Let *x = p*

Then *y = 3x + 1*

So *y = 3p + 1*

There are many ways to write parametric equations, as you can use different parameters other than *p*. We can also change parametric equations back into Cartesian form.

For example,

Find the Cartesian equation of *x** = 3t + 1*, *y = 2t - 3*.

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We use the process for solving simultaneous equations to eliminate the parameter.

## Parametric Representation of a Curve

For the equation of a parabola , the parametric equation can be written as:

where *t* is a parameter.

Therefore, the other types of parabola can be written as below.

Parabola can be written as:

Parabola can be written as:

Parabola can be written as:

### Examples

1. Given the parabola and , find:

a) its Cartesian equation

b) the points on the parabola when

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a)

b)

2. Find the coordinates of the focus and the equation of the directrix of the parabola .

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## Chords, Tangents and Normals

A number of parabola parametric properties are below:

If and are any two points on the parabola , then the chord *PQ* has gradient and equation .

If *PQ* is a focal chord, then *pq* = -1.

The tangent to the parabola at the point has gradient *p* and equation given by .

The tangents to the parabola at points and intersect at the point .

The normal to the curve at point has gradient and equation given by

The normals to the parabola at and intersect at

### Examples

1. Find the equation of the chord joining points where *t*=3 and *t=-2* on the parabola .

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2. Find the equation of the tangent to the parabola at the point .

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Another approach to the equations of tangent, normal and chord of a parabola is derived from points in Cartesian form.

If point lies on the parabola , then the equation of the tangent at *A* is given by:

If point lies on the parabola , then the equation of the normal at *A* is given by:

The equation of the chord of contact *XY* of tangents drawn from external point to the parabola is given by:

### Example

Tangents are drawn from the point to the points *P* and *Q* on the parabola . Find the equation of the chord of contact *PQ* and the coordinates of *P* and *Q*.

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## More Properties of the Parabola

The tangents at the end of a focal chord intersect at right angles on the directrix.

For example,

Points and lie on the parabola .

a) Find the equation of line *PQ*.

b) Show that *PQ* is a focal chord

c) Prove that the tangents at *P* and *Q* intersect at right angles on the directrix.

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a) Equation of *PQ*:

b)

c)

The directrix has equation *y = a*, that is *y = 2*.

The point (-3, 2) lies on the line *y* = 2, which is the directrix.

Another property of the parabola is that any point *P* on a parabola is equally inclined to the axis of the parabola and the focal chord through *P*.