A point moves in the -plane so that it always lies on the circle

MATHEMATICS · Multi-part · AP · Calculus AB · Contextual Applications of Differentiation · Contextual Applications of Differentiation

Problem

A point

P\bigl(x(t),\,y(t)\bigr)

moves in the

xy

-plane so that it always lies on the circle

x^{2}+y^{2}=a^{2}

, where

a

is a positive constant. (i) Prove that

x\,\frac{dx}{dt}+y\,\frac{dy}{dt}=0

at every instant of the motion. (ii) Hence state the relation between

\dfrac{dy}{dt}

and

\dfrac{dx}{dt}

when the point is in the first quadrant. (iii) Explain what happens to the sign and magnitude of

\dfrac{dy}{dt}

as the point approaches

(a,0)

along the circle.