Definition of a Parametric Equation
● Cartesian equation – y = f(x)
● Parametric equation – x = f(t), y = g(t)
● Both x and y are defined in terms of a third variable (usually t or θ )
● Parametric equations can be used for a complicated curve which doesn’t
have a simple cartesian equation
● When sketching a parametric curve, find the x and y coordinates of the
values of the third variable, plot these points and join them
Cartesian Equation of a Parametric Curve
● Need to eliminate the third variable (e.g t or θ )
● Three methods:
○ Make t the subject of the x and y equations and substitute
○ Add/subtract the x and y equations so that it is easy to make t the
subject, then substitute and simplify
○ If the parametric equations involve trigonometric functions, find the
identity which connects the two trigonometric functions and
substitute
● Parametric equations can describe circles using trigonometry:
○ Circle with radius r and centre (0, 0) has parametric equations
x = r cos θ
y = r sin θ
○ Circle with radius r and centre (a, b) has parametric equations
x = a + r cos θ
y = b + r sin θ
Parametric Differentiation
Finding the Gradient of a Parametric Curve
Finding the Equation of the Tangent and Normal to the Curve
● Using the above method, we can find the gradient at any point on a
parametric curve
● Therefore to find the tangent and normal, we proceed as we would if it was
a cartesian curve
● Remember that the gradient of the normal is the negative reciprocal of the
gradient of the tangent
Finding the Turning Point of a Parametric Curve
● Find the gradient function of the parametric curve as per the method
above
● Set this function to 0
● Find the values of t or θ at this point and substitute into the parametric
equations to find the x and y coordinates of the turning points