Question

What is the center and radius of the circle with equation x^2 + y^2 + 18x – 12y – 27 = 0?

Answer 1

Center: (9, –6); Radius = 144

Center: (–9, 6); Radius = 144

Center: (9, –6); Radius = 12

Center: (–9, 6); Radius = 12

Answer 2

Hello, NK!

Please re-arrange x^2 + y^2 + 18x – 12y – 27 = 0 with all the x-terms first and all the y-terms second, and the constant (now +27) on the right side of your equation.

Next move: complete the square. How would you complete the square for x^2 + 18x?

Answer 3

Would x^2 + 18x + y^2 -12y = 27 be my equation?

Answer 4

Actually, I'm not quite sure how I'd complete the square.

Answer 5

Great! You're good at following directions. Now, complete the square of x^2 + 18x.
Have you any reference work available (textbook, online learning materials, etc.)?

Answer 6

I'm sure you're familiar with the general formula for a quadratic function:

ax^2 + bx + c = 0.

Compare that to your

1x^2 + 18x

Comparing coefficients, you see that a=1 and b=18, right?

Answer 7

Yes

Answer 8

1. Multiply your "b" by 1/2.
2. square the result.
3. Add the result to your x^2 + 18 x, and then subtract that same result.

What do you get?

Answer 9

x^2 + 18x + 81 = 0
x^2 + 18x = -81 ?

Answer 10

Make it x^2 + 18x + 81 -18.
Just hold that for now.
Next, look at the y terms: y^2 -12y.

What is "b" in this case?

Answer 11

-12

Answer 12

Cool. and what is the SQUARE of "HALF OF b"?

Answer 13

36?

Answer 14

this is exactly what you did before.

Answer 15

Right. So, you take your y^2 - 12y and add 36, then subtract 36.

Now you'll have x^2 + 18x + 81 - 81 + y^2 and... what else?

Please finish typing your whole equation, including that +27 on the right side.

Answer 16

x^2 + 18x + 81 - 81 + y^2 -12y + 36 - 36 = 27

Would I subtract the 81 & 36 now?

Answer 17

Add 81 to both sides of the equation, and then add 36 to both sides of the equation.
this will leave you with x^2 + 18x + 81 on the left side, plus your y terms. Agreed? what does your equation look like now?

Answer 18

x^2 + 18x + 81 + y^2 - 12y + 36 = 144

Answer 19

Yes. Can you group those as (factored) perfect squares now?

Answer 20

Bill: OpenStudy was down, so nk7 and I were continuing work on this problem thru private messaging.

Answer 21

@nk7 has already typed this in: "A circle, and (-9, 6). And it would represent the radius I believe. Sorry about missing the "^2" part".

Answer 22

ah, sounds good. I didn't see you here, and figured he shouldn't be kept waiting unnecessarily.

Answer 23

Many thanks, Bill!
@nk7: would you mind summarizing your results? "the given equation represents a circle with center at ( , ) and with radius ( )"

Answer 24

the given equation represents a circle with center at (-9 , 6 ) and with radius of 12

Answer 25

Nice work!! Any questions about the "completing the square" process?

Answer 26

None at all, you explained it quite well. I'm a she by the way :) Thank you both immensely. mathmale, thank you for actually explaining this to me rather than just giving me the answer. I actually understand this concept now! And whpalmer4, thank you for stepping in when you thought no one was around. This helped alot!

Answer 27

congratulations on having learned so quickly and done so well here. I also want to thank you for expressing your appreciation. Hope we can work together again in the future!