Higher Engineering Mathematics, Sixth Edition

The circle and its properties 127

11. Determine the lengthof theradius and circum-
    ference of a circle if an arc length of 32.6cm
    subtends an angle of 3.76rad.
    [8.67cm, 54.48cm]


12. Determinetheangleoflap,indegreesandmin-
    utes, if 180mm of a belt drive are in contact
    with a pulley of diameter 250mm.
    [82◦ 30 ′]


13. Determinethenumberofcompleterevolutions
    a motorcycle wheel will make in travelling
    2km, if the wheel’s diameter is 85.1cm.
    [748]


14. The floodlights at a sports ground spread its
    illuminationover an angle of 40◦to a distance
    of 48m. Determine (a) the angle in radians,
    and (b) the maximum area that is floodlit.
    [(a) 0.698rad (b) 804.1m^2 ]


15. Determine (a) the shaded area in Fig. 13.10
    (b) the percentage of the whole sector that the
    area of the shaded portion represents.
    [(a) 396mm^2 (b) 42.24%]

0.75rad 50 mm

12 mm

Figure 13.10

16. Determine the length of steel strip required to
    make the clip shown in Fig. 13.11.
    [701.8mm]

100mm

125mm

rad

100mm

1308

Figure 13.11

17. A 50◦tapered hole is checked with a 40mm
    diameter ball as shown in Fig. 13.12. Deter-
    mine the length shown asx.
    [7.74mm]

70mm

x

508

40mm

Figure 13.12

13.5 The equation of a circle

The simplest equation of a circle, centre at the origin,

radiusr, is given by:

x^2 +y^2 =r^2

For example, Fig. 13.13 shows a circlex^2 +y^2 =9.

More generally, the equation of a circle, centre (a,b),

radiusr, is given by:

(x−a)^2 +(y−b)^2 =r^2 (1)

Figure 13.14 shows a circle(x− 2 )^2 +(y− 3 )^2 =4.

The general equation of a circle is:

x^2 +y^2 + 2 ex+ 2 fy+c=0(2)

3

3

2 x

(^21) y (^259)
x
y
2
1
(^01)
21
21
22
22
23
23
Figure 13.13
Multiplying out the bracketed terms in equation (1)
gives:
x^2 − 2 ax+a^2 +y^2 − 2 by+b^2 =r^2