50 POINT TO WHO ANSWERS IT RIGHT PLZ FOR A FINAL 2. The general form of a circle is given as x2 + y2 + 4x -12 y+4=0.
(a) What are the coordinates of the center of the circle?
(b) What is the length of the radius of the circle?

The

general equation of a circle

is

∣

∣

∣

∣

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

a

a

x

2

+

y

2

+

2

g

x

+

2

f

y

+

c

=

0

a

a

∣

∣

−−−−−−−−−−−−−−−−−−−−−−−−−−−−−

with centre =

∣

∣

∣

∣

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

a

a

(

−

g

−

f

)

a

a

∣

∣

−−−−−−−−−−−−−−

and radius (r) =

∣

∣

∣

∣

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

a

a

√

g

2

+

f

2

−

c

a

a

∣

∣

∣

−−−−−−−−−−−−−−−−

the equation

x

2

+

y

2

+

4

x

−

4

y

−

1

=

0

is in this form

and by comparing the coefficients of like terms we get the values for g ,f and c.

2

g

=

4

⇒

g

=

2

,

2

f

=

−

4

⇒

f

=

−

2

and

c

=

−

1

⇒

centre

=

(

−

g

,

−

f

)

=

(

−

2

,

2

)

and r =

√

g

2

+

f

2

−

c

=

√

2

2

+

(

−

2

)

2

−

(

−

1

)

=

√

9

=

3

Alternatively the method of

completing the square

can be used to obtain the equation.

(

x

2

+

4

x

+

4

)

+

(

y

2

−

4

y

+

4

)

−

1

=

4

+

4

⇒

(

x

+

2

)

2

+

(

y

−

2

)

2

=

9

The

standard form of the equation of a circle

is

∣

∣

∣

∣

¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯

a

a

(

x

−

a

)

2

+

(

y

−

b

)

2

=

r

2

a

a

∣

∣

−−−−−−−−−−−−−−−−−−−−−−−−−

where (a ,b) are the coordinates of the centre and r, the radius.

⇒

centre

=

(

−

2

,

2

)

and

r

=

3

an advantage of the standard deviation is that it increases as the dispersion of the data increases. e. the interquartile range is preferred when the data are not skewed or no have outliers. an advantage of the standard deviation is that it uses all the observations in its computation.