Question

What are the coordinates of the center of the circle shown below?

Answer 1

what have you tried?

Answer 2

nothing yet. I was about to graph it in my calculator and then find the center; that's what I usually do. But I want to try to do it without graphing it.

Answer 3

You have to put it into standard form and you'll recognize the center coordinates almost immediately

Answer 4

Yeah, I figured. But I don't think I'm clear on how to do that.

Answer 5

do you know the standard form of a circle?

Answer 6

isn't it x^2+y^2=r?

Answer 7

hmm it's actually (x-h)^2 + (y-k)^2 = r^2

what you wrote is the standard form of a circle when the center is at origin

Answer 8

oh ok

Answer 9

x^2 + y^2 -8x - 10y - 8 = 0

now is it alright with you if i rewrite it as

x^2 - 8x + y^2 - 10y = 8

Answer 10

yes

Answer 11

now i group them

(x^2 - 8x) + (y^2 - 10y) = 8

nothing fancy here..

Answer 12

Okay

Answer 13

now..do you know how to use completing the square?

Answer 14

I know I learned it a while ago, but I don't really remember it

Answer 15

hmm that's a jiffy

Answer 16

lgba is doing the proper long version ;) i just cut to the halving and be done with it :P

Answer 17

well let me tell you what we're gonna do first..

we have

(x^2 - 8x) + (y^2 - 10y) = 8

we want to turn it into (x-h)^2 + (y-k)^2

that means we need to turn (x^2 - 8x) and (y^2 - 10y) into square of binomial..to do that..we need a perfect square trinomial...to make this into a perfect square trinomial..we need to perform completing the square to get the third term...do you get the objective?

Answer 18

there's a shortcut? maybe you can share it @amistre64

Answer 19

since the completed square amounts to (a+b/2)^2; and we are looking for b/2 ....

Answer 20

the other parts are helpful for finding the radius tho so, it godd to know th elong version

Answer 21

hmm i guess since it's just looking for center that works

Answer 22

I don't understand...

Answer 23

if you dont understand, then you really need to become adept at the method lgba is proscribing

Answer 24

nice typo lol, proscribe means to ban or condemn lol

Answer 25

okay let me demonstrate the completing the square method

let's say we have (x^2 + 4x) i want to turn this into a perfect square trinomial

my first step is to divide the coefficient of "x" by two...in this case the coefficient of x is 4 (because of 4x) so i divide it by 2. 4/2 = 2

now the next step is to square it. 2^2 = 4

so to make x^2 + 4x into a perfect square trinomial i need to add 4

x^2 + 4x + 4<--now it's a perfect square trinomial

Answer 26

do you get the demo?

Answer 27

I think so.. but if you add four, don't you also have to add four to the other side of the equation or is it different in this case?

Answer 28

very good!! yes you do

Answer 29

so we have

(x^2 - 8x) + (y^2 - 10y) = 8

try to complete the square in (x^2 - 8x)

Answer 30

umm would it be (x^2 - 8x + 16)

Answer 31

correct..like you said add 16 to the other side as well

(x^2 - 8x + 16) + (y^2 - 10y) = 8 + 16

(x^2 - 8x + 16) + (y^2 - 10y) = 24

you get htat right?

Answer 32

yes. ok so (y^2-10y) would become (y^2-10y +25). Then (x^2 - 8x + 16) + (y^2 - 10y + 25) = 49

Answer 33

correct!!!

Answer 34

woo hoo :)

Answer 35

so now you turn (x^2 - 8x + 16) into a square of a binomial

Answer 36

(x-4)^2

Answer 37

right! and (y^2 - 10y +25)?

Answer 38

(y-5)^2

Answer 39

congratulations you just turned it into standard form!

(x - 4)^2 + (y - 5)^2 = 49

now can you determine the center?

Answer 40

The center is (h,k) so (4,5).

Answer 41

correct!

Answer 42

Thank you soooo much!! :)

Answer 43

you're welcome!!!