Mathematics

Problem

Let

C

be the circle

x^{2}+y^{2}-2ax=0

where

a>0

. A point

P

C

can be described by the parameter

\theta

through the relations

x=a(1+\cos\theta),\; y=a\sin\theta\;(0<\theta<2\pi)

. a. (i) Verify that every point

P

obtained from the above relations satisfies the equation of

C

. b. (ii) Find the slope of the tangent to

C

at the general point

P

in terms of

\theta

. c. (iii) Hence write down the slope and the equation of the normal at

P

. d. (iv) For

\theta=\dfrac{\pi}{3}

, find the coordinates of

P

and obtain the explicit equations of the tangent and the normal at that point.