Basic Engineering Mathematics, Fifth Edition

236 Basic Engineering Mathematics

19. Determine the length of steel strip required
    to make the clip shown in Figure 26.13.

125mm

rad

130 

100mm

100mm

Figure 26.13

20. A 50◦tapered hole is checked with a 40mm
    diameter ball as shown in Figure 26.14.
    Determine the length shown asx.
    70mm
    x

50 

40mm

Figure 26.14

26.5 The equation of a circle

The simplest equation of a circle, centre at the origin

and radiusr,isgivenby

x^2 +y^2 =r^2

For example, Figure 26.15 shows a circlex^2 +y^2 =9.

3

3

2

x^2 y^2  9

x

y

2

1

0 1

 1

 1

 2

 2

 3

 3

Figure 26.15

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

and radiusr,isgivenby

(x−a)^2 +(y−b)^2 =r^2 (1)

Figure 26.16 shows a circle(x− 2 )^2 +(y− 3 )^2 =4.

r^5

2

y

5

4

2

024 x

b 53

a 52

Figure 26.16

The general equation of a circle is

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

Multiplying out the bracketed terms in equation (1)

gives

x^2 − 2 ax+a^2 +y^2 − 2 by+b^2 =r^2

Comparing this with equation (2) gives

2 e=− 2 a,i.e.a=−

2 e

2

and 2 f=− 2 b,i.e.b=−

2 f

2

and c=a^2 +b^2 −r^2 ,i.e.r=

√

a^2 +b^2 −c

Thus, for example, the equation

x^2 +y^2 − 4 x− 6 y+ 9 = 0

represents a circle with centre,

a=−

(

− 4

2

)

,b=−

(

− 6

2

)

i.e., at (2, 3) and

radius, r=

√

22 + 32 − 9 = 2

Hence,x^2 +y^2 − 4 x− 6 y+ 9 =0isthecircleshownin

Figure26.16 (which may bechecked by multiplyingout

the brackets in the equation(x− 2 )^2 +(y− 3 )^2 = 4 ).

Problem 18. Determine (a) the radius and (b) the

co-ordinates of the centre of the circle given by the

equationx^2 +y^2 + 8 x− 2 y+ 8 = 0