Question

Identify the center and radius of each

1. 364+28y+y^2+x^2= -26x

2. x^2+ y^2+ 24x+ 10y+ 160= 0

3. -6x= -x^2+ 32y- 264- y^2

4. -6x+ x^2= 97+10y- y^2

Answer 1

The typical formula for a circle is (x-x1)^2+(y-y1)^2=r^2 where (x1,y1) is the center and r is the radius. So the first thing you want to do is get all of those messy equations into the circle format. Looks like this is a practice for Completing the Square (CTS)

#2 looks nice :)
x^2+y^2+24x+10y+160=0
first group the x's and the y's together so that we can factor. Might as well subtract 160 from each side too...
x^2+24x+y^2+10y=-160
(x^2+24x)+(y^2+10y)=-160

For the x's
x^2+24x -> we want to Complete the Square! (CTS)
To complete the square, you take 1/2*b (1/2 of the second term's coefficient) and square it!
24x is the second term, 24 is the coefficient, 24=b
1/2 of 24 = 12
12^2 is 144
Then we get (x^2+24x+144)-144 (don't forget to subtract this out... we are trying to "add zero" creatively!)
We can factor this into (x+12)^2 -144

For the y's
y^2+10y -> CTS!
10y is the second term, 10 is the coefficient
1/2 of 10 = 5
5^2 = 25
Then we get (y^2+10y+25) -25
We can factor this into (y+5)^2 - 25

Putting this all together:
(x+12)^2 - 144 + (y+5)^2 - 25 = -160
Simplify by adding 144 and 25 to both sides...
(x+12)^2 + (y+5)^2 = 9
so then we know the center is (-12, -5) and the radius is sqrt(9)=3
Remember to use the opposite signs in the parenthesis when finding the center, and to take the square root of the right side to find the radius!

Good luck with the other four! HTH!