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.