Question

Find the center and radius of the circle whose equation is x^2−1x+y^2−7y−14=0.

Answer 1

You must get this idea in your head: \((a+b)^{2} = a^{2} + 2ab + b^{2}\). The really important parts are anything with a 'b' in it. This is what we leverage to solve the problem.

Answer 2

Once we identify 'x' in our problem statement with 'a' in our fundamental idea, we can focus on the values that correspond to 'b'.

\(x^{2} - x\) -- We need to identify -1 as \(2b\). Given \(2b\), how do we find \(b^{2}\)?

Answer 3

is it y?

Answer 4

tkhunny, are you there?

Answer 5

"y"? Where did that come from? We're only talking about 'x'.

Answer 6

It is an independent question. Put everything else out of your mind.

If \(2b = -1\), what is \(b^{2}\)?

Answer 7

.25

Answer 8

\((-1/2)^{2} = 1/4\) - Perfect. Now to our problem statement...

x^2−1x+y^2−7y−14=0

Thinking about just he x-parts, we already have the solution we need.

x^2 − x + 1/4 + y^2 − 7y − 14 = 0

What do you thnk about that?

Answer 9

looks similar to what i have here

Answer 10

Well that's good and bad. It is TOWARD what we want, but as stated it is horribly wrong! We can't just add 1/4 and have the same problem! We have to fix that.

x^2 − x + 1/4 + y^2 − 7y − 14 = 1/4

Okay, now I feel better. We added 1/4 to each side and retained the equality. We calculated 1/4 as what we needed, and then just copied it on the other side so the world would not end.

Awesome. Now, you do the same thing for the y-values. What shall we add to both sides so that the y-parts make a perfect square trinomial?

Answer 11

-14

Answer 12

Bzzzzzt! (That was the bad buzzer!)

You didn't ask the right question.

If \(2b = -7\), what is \(b^{2}\). It is \(b^{2}\) that we need.

Answer 13

12.15

Answer 14

12.25, I hope you mean. \(\left(\dfrac{7}{2}\right)^{2} = \dfrac{49}{4}\)

Now we have:

x^2 − x + 1/4 + y^2 − 7y + 49/4 − 14 = 1/4 + 49/4

What's next? I'm pretty excited. I think we're close!

Answer 15

do we have to factor and simplify?

Answer 16

Sort of. Why did we calculate those goofy little numbers? What was our purpose in doing that? (Please refer to the fundamental idea where we started this conversation.)

Answer 17

determine b? im really struggling with this sorry

Answer 18

No worries. There's a lot going on.

We calculated those little numbers for the sole purpose of creating a perfect square trinomial. Given that, we can rewrite as a squared binomial.

x^2 − x + 1/4 + y^2 − 7y + 49/4 − 14 = 1/4 + 49/4

Rewrite

\((x - 1/2)^{2} + (y - 7/2)^{2} - 14 = 1/4 + 49/4\)

Does that look better?

Answer 19

yes

Answer 20

i figured it out thanks

Answer 21

What did you get? Hopefully, there was some simplifying going on, there.

Answer 22

center of circle is 1/2, 3.5
radius is 5.14

im going to post a new question ok please look at it thanks

Answer 23

Ugly decimals. But GREAT work, otherwise.