Solving Circles?

Question

The question is :
given the circle X2+y2+4x-6y+12=0. Find the coordinates of its center (h,k) and its radius r.

Answer 1

x^2 + y^2 + 4x - 6y + 12 = 0
x^2 + 4x + y^2 - 6y + 12 = 0
(x^2 + 4x) + (y^2 - 6y) + 12 = 0
(x^2 + 4x + 4 - 4) + (y^2 - 6y + 9 - 9) + 12 = 0
((x^2 + 4x + 4) - 4) + ((y^2 - 6y + 9) - 9) + 12 = 0
((x + 2)^2 - 4) + ((y - 3)^2 - 9) + 12 = 0
(x + 2)^2 - 4 + (y - 3)^2 - 9 + 12 = 0
(x + 2)^2 + (y - 3)^2 - 1 = 0
(x + 2)^2 + (y - 3)^2 = 1

Circle Equation
(x - h)^2 + (y - k)^2 = r^2
r = radius
(h,k) equals center

ANS :

center = (-2,3)
radius = 1

Answer 2

This involves completing the square:


x^2 + y^2 + 4x - 6y + 12 = 0

x^2 + 4x + y^2 - 6y + 12 = 0

(x^2 + 4x) + (y^2 - 6y) + 12 = 0

(x^2 + 4x) + (y^2 - 6y) = -12

(x^2 + 4x + 4) + (y^2 - 6y + 9) = -12 + 4 + 9

(x + 2)^2 + (y - 3)^2 = 1

Now you are in standard form for a circle:

(x - h)^2 + (y - k)^2 = r^2

so, h = -2 and k = 3 and r = 1
center: (-2,3) r = 1

Answer 3

equation of a circle = (x-h)^2 + (y-k)^2 = r^2
to get it into this form:
First rearrange & group
(x^2 + 4x) + (y^2 - 6y) = -12
In the first group, divide 4 (because of 4x) by 2 and square (4/2 = 2, 2^2 = 4). Add the result to the end of the group and also add it to the other side.
(x^2 + 4x + 4) + (y^2 - 6y) = -12 + 4
For the second group, do similar. Divide 6 (because of 6y) by 2 and square (6/2 = 3, 3^2 = 9). Add it as you did with the 1st group
(x^2 + 4x +4) + (y^2 - 6y +9) = -12 +4 + 9
Factor
(x + 2)^2 + (y - 3)^2 = 1
so the center is (-2, 3) with a radius of one.

the more you do this, the quicker you'll get and can combine some of the steps

Answer 4

home work?

first convert the given eqn into (x-a)^2+(y-b)^2=r^2 form

so we have (x^2+4x+4) + (y^2-6y+9) = 1
or, (x+2)^2 + (y-3)^2 = 1

so the center must be (-2,3) and the radius = 1

there is a general solution.... easy to remember... refer to your textbook!

Answer 5

compare the given by the expanded formula
(x-a)^2+(y-b)^2=r^2
or
x^2+y^2-2ax-2by+a^2+b^2-r^2=0
Thus
-2a=4
a=-2

-2b=-6
b=3

a^2+b^2-r^2=12
4+9-r^2=12
13-12=r^2
r=1

Answer:
center (h,k)=(-2,3)
r=1