# Parametric Equations

## Understanding Parametric Equations

- In
**coordinate geometry**, an equation can be represented parametrically, this means using another variable (the parameter) to define the variables within the equation. - A
**parametric equation**expresses the coordinates x and y in terms of one parameter, traditionally denoted as ‘t.’ The expressions x(t) and y(t) define the x and y coordinates as functions of t. - For example, a circle’s parametric equations are x = rcos(t), y = rsin(t). Here, r is the radius and t is the angle subtended at the centre of the circle by the line from the origin to the point (x,y).

## Drawing Curves from Parametric Equations

- To sketch curves defined by parametric equations, substitute a range of values for the parameter t into both x(t) and y(t) to generate coordinates for points on the curve.
- When sketching
**parametric equations**, it’s essential to consider the range of t for which the functions are defined.

## Converting Between Parametric and Cartesian Forms

- A parametric representation can be converted into the
**Cartesian form**and vice versa. This involves eliminating the parameter. - To convert from parametric to Cartesian form, express t in terms of x or y in one equation, and then substitute this expression into the other.

## Differentiating and Integrating Parametric Equations

- The derivative dy/dx can be found by using the Chain Rule. This involves finding dx/dt, dy/dt and then applying the formula dy/dx = (dy/dt) / (dx/dt).
- When it comes to
**integration**, we can find the area under a curve defined by parametric equations by manipulating the form ∫y dx into a form in terms of t.

## Application of Parametric Equations

**Parametric equations**are commonly used in physics to describe the motion of objects, where ‘t’ is time. The x and y coordinates represent the object’s position at the given time t.- They enable the simple representation of more complex shapes, like the flight path of a projectile or the shape of a rollercoaster loop, and are utilized in the technology behind computer graphics and animations.