### Parametric Equations

When the coordinates of curve are expressed in terms of a third independent variable - parameter - parametric equations are produced.

**E.g.**

y = 2t; x = t^{2} - 3

Here, 't' is the parameter.

**E.g.1**

y = t^{2} + 2; x = (t-3);

Now, we have a choice - finding the values of x and y manually by substituting the values of t or use algebra to find the *Cartesian equation. Let's go for the latter.*

t = (x+3) => y =(x + 3)^{2} + 2

This can easily be sketched: basic quadratic curve translated -3 in the x-axis and 2 in the y axis.

**E.g.2**

y = t; x = (t - 3)^{2};

t = √x+3 => y = √x + 3

This can easily be sketched as follows:

**E.g.3**

y = sec t; x = cos t

xy = cos t . 1/cos t

xy = 1

y = 1/x
This is reciprocal curve.

This can easily be sketched as follows:

**E.g.4**

y = 3 sin t; x = 3 cos t

cos t = x/3; sin t = y/3;

x^{2} / 9 + y^{2} / 9 = sin^{2} t + cos^{2} t = 1

x^{2} / 9 + y^{2} / 9 = 1

x^{2} + y^{2} = 9

x^{2} + y^{2} = 3^{2}

This is a circle with radius 3.

This can easily be sketched as follows:

**E.g.5**

y = -2 + 3 sin t; x = 3 + 3 cos t

cos t = (x-3)/3; sin t = (y+2)/3;

(x-3)^{2} / 9 + (y+2)^{2} / 9 = sin^{2} t + cos^{2} t = 1

(x-3)^{2} / 9 + (y+2)^{2} / 9 = 1

(x-3)sup>2 + (y+2)^{2} = 9

(x-3)^{2} + (y+2)^{2} = 3^{2}

This is a circle with the centre at (3,-2) and radius 3.

This can easily be sketched as follows:

**E.g.6**

y = 3 sin t; x = 4 cos t

cos t = x/4; sin t = y)/3;

x^{2}/16 + y^{2} / 9 = sin^{2} t + cos^{2} t = 1

x^{2} / 16 + y+^{2} / 9 = 1

if x = 0 => y = +3 or -3

if y = 0 => x = +4 or -4

It is an ellipse.
This can easily be sketched as follows:

#### The World of Spirals

The significance of parametric equations can be seen from the following beautiful shapes, which are produced by the manipulation of them in different ways.

Please press *clear button*, before choosing a new spiral, so that you can see each smoothly.

#### Parametric Equations - interactive practice

You can enter the parametric equations in terms of t in the two text boxes and then press the *Enter* key. After that move the slider to see the figure being built up on the Cartesian plane.

#### Area under a curve

Suppose we want to find the area under the curve, y = f(t), using parameter, t in the range, a<=x<=b. x = g(t).

Area = _{a}∫^{b}y dx

Since ∫ y dx = y (dx/dt) dt

Area = _{a'}∫^{b'}f(t) g'(t) dt

a' and b' are the boundaries in terms of t.

#### Area of a Circle

The parametric equations for the circle of radius r are as follows:

y = r sin (t); x = r cos(t); dx/dt = -r sin(t)

Area = _{0}∫^{2π}y (dx/dt) dt

= _{0}∫^{2π}r sin(t). -r sin (t) dt

= _{0}∫^{2π}-r^{2} sin^{2}(t) dt

= _{0}∫^{2π}-r^{2} (1- cos(2t))/2 dt

= _{0}∫^{2π}-r^{2}/2 (1- cos(2t)) dt

= -r^{2}/2 [t - sin(2t)/2]^{2π}_{0}

= -r^{2}/2 [2π]

= -πr^{2}

Area of a circle = πr^{2}

#### Area of an Ellipse

The parametric equations for the ellipse of semi-major axis and semi-minor axis a and b respectively are as follows:

y = a sin (t); x = b cos(t); dx/dt = -b sin(t)

Area = _{0}∫^{2π}y (dx/dt) dt

= _{0}∫^{2π}a sin(t). -b sin (t) dt

= _{0}∫^{2π}-ab sin^{2}(t) dt

= _{0}∫^{2π}-ab (1- cos(2t))/2 dt

= _{0}∫^{2π}-ab/2 (1- cos(2t)) dt

= -ab/2 [t - sin(2t)/2]^{2π}_{0}

= -ab/2 [2π]

= -πab

Area of an ellipse = πab

If a = b = r, it is a special case - a circle.

Area = πr^{2}

#### Creating a Heart - interactive

In the following applet, two parametric equations involving trigonometry create a heart. The parameter changes from 0 radians to 2π radians. Please click the *play* button at the bottom.