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

25

step-by-step explanation:

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step-by-step explanation: