The circle C has equation x^2 + y^2 - 6x + 4y = 12. (a) Find the centre and the radius of C.

Step 1: Note down the equation and convert that equation into the standard form of the circle
NOTE: Standard form (x - h)^2 + (y - k)^2 = r^2
Where center (h, k) and radius r
Skill 1: Take down the equation and separate the similar variables.
EXAMPLE: x^2 + y^2 - 6x + 4y = 12
(x^2 - 6x) +(y^2 + 4y) = 12
Skill 2: According signs of the equation convert these as
either (a+b)^2 \text{ or } (a-b)^2.
Skill 3: Find out the a and b by convert the equation to
a^2 ± 2ab and compare.
EXAMPLE: [/math]x^2 - 6x[/math]
a = x
for b =?, take middle term - 6x = 2*3*x
And compare it with the 2ab
b = 3.
Skill 4: Substitute the values of a and b in the formulas of (a±b)^2.
NOTE: Use the formulas of (a+b)^2 \text{ or } (a-b)^2
(a+b)^2 = a^2 + 2ab + b^2.
a^2+2ab=(a+b)^2-b^2.
or
a^2 - 2ab = (a-b)^2 - b^2 .
EXAMPLE:
[math][(x-3)^2-9]+\left[\left(y+2\right)^2-4)\right]=12[/math]
(x-3)^2+(y+2)^2=12+9+4
(x-3)^2+(y+2)^2=25
Step 2: Note down the given circle equation
EXAMPLE: (x -2)^2 + (y + 1)^2 = \frac{169}{4}
Step 3: Compare the given equation with the Standard Form of circle equation.
EXAMPLE: [math](x-2)^2+\left[y-\left(-1\right)\right]^2=\frac{169}{4}[/math] Compare it with
(x - h)^2 + (y - k)^2 = r^2
Where center (h, k) and radius r
Step 4: Identify the r, h and k values.
NOTE: r- radius
h is x-coordinate
k is y- coordinate of the center.
EXAMPLE; (h, k) = (2, -1), r = \frac{168}{4}