Understanding Engineering Mathematics

represent a circle with centre (a,b) and radiusr, since:

(x−a)^2 +(y−b)^2 =r^2

The parameterθ can be regarded as the angle made by the radius

with thex-axis, as shown in Figure 7.10.

y

0 x

r

(a,b)

q

(a + r cos q, b + r sin q)

Figure 7.10Parametric form of a circle.

So, forx=2cost−1andy=2sintwe get

2cost=x+ 1

so

(x+ 1 )^2 +y^2 =4cos^2 t+4sin^2 t= 4

giving a circle centre (−1, 0) and radius 2.

(iv)x=cos 2t,y=cost

So from the double angle formula (188

➤

)

x=cos 2t=2cos^2 t− 1 = 2 y^2 − 1

which we may as well leave in the implicit form

x= 2 y^2 − 1

7.3 Reinforcement

7.3.1 Coordinate systems in a plane

➤➤

204 205

➤

A.Plot the points

(i) (−1,−1) (ii) (3, 2) (iii) (−2, 3) (iv) (0, 4)

(v) (4, 0) (vi) (1, 1) (vii) (3,−1) (viii) (0,−2)