Finding the Equations of Tangents and Normals to Parametric Curves

Parametric curves take the formwhereis a parameter. Each value of gives (not necessarily unique) values ofandIf we need to find the equation of a tangent or normal to the curve at any pointthen we need the gradientat that point. We findusing the identitywill be a function ofso if we have the vale ofwe can findthe coordinatesthen useto find the equation of the tangent orto find the equation of the normal.

Example: Find the equation of the tangent and normal for the curve given in parametric coordinates asat

At

Tangent:

Normal:

If we have the pointbut notthen we have to use the point to find

Example: Find the equation of the tangent and normal for the curve given in parametric coordinates asat the point

We have to find the value ofandFrom these two,

Tangent:

Normal: