Let a, b E R and r E E41 Consider a circle centred at (a, b) with radius r. We lmow the solutions to the equation (a: a): + {y ii}? = 1'2 describes the circle. (a) Give mo differentiable flmctions f, y : R } E which satisfy: i. a: 6 1s, {t} a}? + (go) 532 = r2. ii.'1:r'1:,y E R, [F [acas}2 + [y b)2 = r2,TI-I_EN Eli E Est. fit) = m,g{t) = 3;. iii. 'u't E R, f'{t) 7E G or g'(t) 3% I]. Hint: Use 3. This is a pnrnmetsrizntion of the curve (1: o)2 + {y in)2 = r2. {f{t},g{t}} can be visualized as tracing out the circle as t varies. [b] Let {mmy} be a point on the circle. Compute % for [I {IJZ + [y in}? = r2 at [amp-u} using implicit differentiation. State where your computation is valid. (c) Let in E JR. Compute {5% for [.12 a): + {y b}: = r2 at {f{t),g[tu)} using the chain rule between the quantities [y = g, :1: = f, and t}. State where your computation is valid. (d) Let to E R. Conrm that at [fritu], Maul}, your results in (b) and (c) are the same. [e] For what values of k E JR is the line 3; = in: + 1 tangent to the circle with centre (5, 1) and radius 4