# 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.