Consider a plane curve defined parametrically by

MATHEMATICS · Multi-part · AP · AP Precalculus · Functions with Parameters, Vectors, and Matrices · Functions with Parameters, Vectors, and Matrices

Problem

x=f(t),\;y=g(t)

, where

f(t)

and

g(t)

are differentiable functions. (i) Define the tangent line and the normal line to the curve at the point corresponding to a parameter value

t=t_{0}

. (ii) Prove that, provided

f'(t_{0})\neq 0

, the slopes of the tangent and the normal lines at

t=t_{0}

are negative reciprocals of each other. (iii) For the curve

x=a\cos t,\;y=a\sin t\;(a>0)

, show that the tangent at any parameter value

t_{0}

is perpendicular to the radius vector drawn from the origin to the point

\bigl(a\cos t_{0},\,a\sin t_{0}\bigr)