Question

An equation of a circle is x2 + y2 + 10x + 6y + 18 = 0. Show all your work in determining the center and radius of this circle. In complete sentences, explain the procedure used.

Answer 1

@jim_thompson5910

Answer 2

@aum

Answer 3

we have

x^2 and 10x

what else is needed to make it a perfect square?

Answer 4

x^2 + 10x + ______

Answer 5

10x ?

Answer 6

why 10x ?

Answer 7

it's in the original equation

Answer 8

I'm pulling all the x terms together

Answer 9

oh okay, so we are taking x^2+10x. Would we bring y^2 @jim_thompson5910

Answer 10

don't worry about the y terms right now

Answer 11

oh okay

Answer 12

what goes in the blank

x^2 + 10x + ______

to make that a perfect square trinomial

Answer 13

+18

Answer 14

try again

Answer 15

+0

Answer 16

is it from the equation?

Answer 17

Focus on 10x

what's half of 10?

Answer 18

5. But can't it also be 25 ?

Answer 19

then square 5 to get 25

so x^2 + 10x + 25 is a perfect square

it factors to (x+5)^2

Answer 20

yes

Answer 21

So add 25 to both sides of the original equation to get


x^2 + y^2 + 10x + 6y + 18 = 0

x^2 + y^2 + 10x + 6y + 18 + 25 = 0+25

(x^2+ 10x + 25) + y^2 + 6y + 18 = 25

(x+5)^2 + y^2 + 6y + 18 = 25

Answer 22

you must do the same for the y terms as well

Answer 23

oh that makes sense so far.

Answer 24

y^2 + 6y + _____

Answer 25

9

Answer 26

I just cut 6y in half and squared it.

Answer 27

add 9 to both sides of

(x+5)^2 + y^2 + 6y + 18 = 25

Answer 28

and complete the square for y to factor

Answer 29

(x+5)^2 + y^2 + 6y + 18+9 = 25+9

Answer 30

keep going

Answer 31

okay so:
(x+5)^2 + y^2 + 6y +27=35
x^2+10x+25+ y^2 + 6y +27=35
x^2+ y^2+10x+ 6y +52=35

Did i do it right?

Answer 32

don't combine 18+9 to get 27

Answer 33

move that +9 over with the y^2+6y

Answer 34

you're going to factor y^2+6y+9 to get (y+3)^2

Answer 35

also, 25+9 = 34, not 35

Answer 36

wait I am so confused, can you show me :/

Answer 37

(x+5)^2 + y^2 + 6y + 18+9 = 25+9

(x+5)^2 + y^2 + 6y+9 + 18 = 25+9

(x+5)^2 + (y^2 + 6y+9) + 18 = 25+9

(x+5)^2 + (y+3)^2 + 18 = 34

Answer 38

you still have more work to do though

Answer 39

okay

Answer 40

You have to get the equation into the form

\[\Large (x-h)^2 + (y-k)^2 = r^2\]

Answer 41

so subtract 18 from both sides, and square it?

Answer 42

yes to the first part, no to the second

Answer 43

(x+5)^2 + (y+3)^2 + 18 = 34

(x+5)^2 + (y+3)^2 + 18-18 = 34-18

(x+5)^2 + (y+3)^2 = 16

Answer 44

what's next

Answer 45

I think take the square root of 16 ?

Answer 46

yep sqrt(16) = 4

Answer 47

so 4^2 = 16

Answer 48

r^2 = 4^2 = 16

therefore, r = 4

the radius is 4

Answer 49

ohh makes sense.

Answer 50

what's the center?

Answer 51

umm.(-5,-3)...but I have no clue I got that

Answer 52

(x+5)^2 + (y+3)^2 = 16

(x+5)^2 + (y+3)^2 = 4^2

compare to

\[\Large (x-h)^2 + (y-k)^2 = r^2\]

Answer 53

wait so it's (-5,-3) ?

Answer 54

do i set them to zero?

Answer 55

how do you get (x+5)^2 into the form (x-h)^2

Answer 56

i don't know :(

Answer 57

notice we have x PLUS 5

and we want a MINUS h

Answer 58

so

x+5 = x - (-5)

now that's in the form x-h

Answer 59

h = -5

Answer 60

so we find y-k the same way ?

Answer 61

yes

Answer 62

oh okay

Answer 63

k = ???

Answer 64

-3

Answer 65

y+3=y-(-3)

y=-3

Right? @jim_thompson5910

Answer 66

k = -3

Answer 67

h = -5
k = -3

Answer 68

(h,k) is the center

Answer 69

yes, thank you for your time, I'm sorry if I took up ur time :(.

Answer 70

you didn't and you're welcome