Finding the Equations of Tangents and Normals to Parametric Curves
Parametric curves take the form
where
is a parameter. Each value of gives (not necessarily unique) values of
and
If we need to find the equation of a tangent or normal to the curve at any point
then we need the gradient
at that point. We find
using the identity
will be a function ofso if we have the vale ofwe can find
the coordinatesthen use
to find the equation of the tangent or
to find the equation of the normal.
Example: Find the equation of the tangent and normal for the curve given in parametric coordinates as
at
At
Tangent:
Normal:
If we have the point
but not
then we have to use the point to find
Example: Find the equation of the tangent and normal for the curve given in parametric coordinates as
at the point
We have to find the value of
and
From these two,
Tangent:
Normal:
