Question

I need help with a calculus problem. The equation for the general form of the circle is 2x^2 +8x +2y^2 -y -2 = 0 . I have found the center (h,k) to equal (-2,1/4) but I can not figure out the radius. Can someone please tell me what the radius is and how to get to it

Answer 1

@ganeshie8 is there anyway you could help me?

Answer 2

So first, can you tell me the standard eq for a circle?

Answer 3

\[2x ^{2} +8x +2y ^{2} -y -2\]

Answer 4

In terms of h,k,r

Answer 5

standard form is\[(x-h)^{2} + (y-k)^{2} = r ^{2}\]

Answer 6

alright good, so first step, tell me how you found the center

Answer 7

what did you do?

Answer 8

h=-2 and k=1/4 is what I got for completing the squares and getting it into standard form and that is right. I was told my radius was wrong. I have tried 13 times and gotten different answers each time. Im on a program that's practice problems for calculus and they wont tell me why its wrong.

Answer 9

ok, so walk me through what you did so I can help you figure out the mistake

Answer 10

give me a minute to type it up on here my notes are a disaster

Answer 11

alright, you can do multiple things if you want

Answer 12

okay, in all honesty, I started this problem earlier and stop for a while and moved on because I was getting frustrated. now I cant find the notes, but what I got was correct, I just cant remember how I got those answers, im drawing a complete blank.

Answer 13

Graph it, and then the radius is the point smack in the middle

Answer 14

wouldn't that be the center point not the radius?

Answer 15

and I can't graph it without the radius

Answer 16

Yes you can, I just did. When you find the center, you'll find the radius

Answer 17

ok, so one, calculators are not allowed in most colleges, I strongly recommend learning the written approach 2 just verbalize and tell me for now the process and we can do it together

Answer 18

I don't believe that for a second. It's good to know the written way, but I simply don't believe that calculators aren't allowed.
This might help you visualize.

Answer 19

I was not allowed a calculator past 12th grade and I am now considering a phd in math. It's true. We are not allowed calculators

Answer 20

we have to take all pieces with variables on the left side, get numbers on the right. Then complete the two squares on the lefthand side for x variables and then y variables. then factor and you have the two squares that make h=-2, and k=1/4

Answer 21

alright, now that is one way, good, but my question is what does that give you on the right where you have your numbers?

Answer 22

I had -1, but I don't think that is right and im not quite sure how I got that

Answer 23

ok, so you are probably right

Answer 24

it probably shouldn't be negative one, but let's generalize one step further. What should the -1 represent?

Answer 25

(in theory)

Answer 26

radius?

Answer 27

there is the key, it represents the radius squared

Answer 28

so the reason -1 is weird is because you then have an imaginary radius (\(\sqrt{-1}=i\))

Answer 29

but that cant be right. I tried showing it was the squareroot of -1 and it said that's impossible, then I tried putting in "I" but it wont allow me

Answer 30

right, probably because when you completed the square, you forgot to move the extra numbers to the right side

Answer 31

I know that the number on the right side represents radius squared and to take the square root of it but I keep getting wrong answers, ive tried multiple times, can you please show me what you get for radius squared

Answer 32

so let's think, half the work is done for us since we know the h,k values

Answer 33

we need to first look at the completing the square bit

Answer 34

\[2x^2+8x+2y^2−y−2=0\]
This is our given correct?

Answer 35

yes

Answer 36

alright so my first step is to get rid of the two's on the x^2 and y^2. Can you think of why?

Answer 37

because it wouldn't be possible to complete the square. It would make it this,\[2(x ^{2} +8x) 2(x ^{2}-y) = 2(-1)\]

Answer 38

first part correct equation not quite but you got the idea

Answer 39

so, in order to rid myself of the 2's. I am going to multiply both sides by _____ to get
\(x^2+4x+y2−\frac{1}{2}y−1=0\)

Answer 40

-2

Answer 41

no, we are going to multiply by \(\frac{1}{2}\) can you tell me why?

Answer 42

and remember we do the same thing to both sides

Answer 43

to get rid of the two, its pretty much the same as dividing it by two

Answer 44

yea, can you show me how to do that though, in an eq so I know you get the concept

Answer 45

what do you mean show you how to do that, you already did it up there?

Answer 46

Can you do it step by step? I skipped two steps

Answer 47

which steps did you skip because I don't see them. to be able to complete the square we had to get rid of the 2's therefore you multiplied both sides to get the equation you got above. there was nothing skipped?

Answer 48

yea, I skipped the actual multiplication steps. Those are what I need to see before we continue

Answer 49

so first step write in the mult. second simplify third simplify more (now that I think there are actually 3 steps)

Answer 50

\[2x ^{2}(1/2) +8x(1/2) +2y ^{2}(1/2) -y(1/2) -2(1/2) = x ^{2}+4x +y ^{2} - 1/2y -1\]

Answer 51

ok so I'm gonna call you on a technicality that will get you into trouble later, what does that all equal?

Answer 52

\[x ^{2}+4x +y ^{2}-1/2y = 1\]

Answer 53

yes and no, not what I was fishing for but correct. So you cannot forget to write :
\[2x2(1/2)+8x(1/2)+2y2(1/2)−y(1/2)−2(1/2)=\color{red}{0\circ\frac{1}{2}}\]
otherwise it is incorrect

Answer 54

it seems silly but trust me, teachers bust your you know what for those small inconsistencies

Answer 55

okay, understandable

Answer 56

alright so now that I know you can do the multiplication let's do the complete the evil square that we all detest. So key thing, keep your eq =0 to start. it makes life easier in the end

Answer 57

We now have:
\[x^2+4x+y^2−1/2y-1=0\]

Answer 58

so start with the y's we know we need a -1/4 right?

Answer 59

\[x ^{2} +4x +2 \] and \[y ^{2} -1/2y +1/8\]

Answer 60

uhm yea, that would be what we need, but now look at non-variable number that we have vs the ones we need

Answer 61

we have -1 but we need a 1/8 and a *correction* 4

Answer 62

so we need to add zero in order to have those numbers, how can we add zero?

Answer 63

I don't quite understand what you're saying,

Answer 64

so this is a trick that we need to use. You sort of just have to memorize it's applications so you can recognize when you need it. what is \[\frac{1}{8}+\frac{-1}{8}=?\]

Answer 65

0

Answer 66

yup so if we add that and the respective for 4 to both sides, we don't mess with our nswer because 0+anything =anything right?

Answer 67

yes

Answer 68

alright, so what we shall do is just what I said above, let's show it in math speak, then you can finish the problem for me, ok?

Answer 69

okay

Answer 70

\[(x^2+4x+4)-4+(y^2−1/2y+\frac{1}{8})-\frac{1}{8}-1=0\]

Answer 71

is the math speak for what I just said. To make it easier, I threw in some parenntheses since addition is commutative

Answer 72

and associative

Answer 73

\[(x+2)^{2} + (y-1/4)^{2} = 1+4+1/8 = 5and1/8\]

Answer 74

yup, now what does r=?

Answer 75

\[\sqrt{41/8}\]

Answer 76

mhmm

Answer 77

do you understand?

Answer 78

Yes, i understand but my program says this is wrong

Answer 79

@Destinymasha this is the question if you got my message and have the time

Answer 80

can someone please check the answer we have gotten above. the program i am using says that our radius is incorrect. if you could tell me why and how o get to the correct answer, that would be amazing! i really just want to understand why i am getting this wrong everytime i try.

Answer 81

@Hero could you possibly take a look

Answer 82

@SolomonZelman could you maybe take a look at this?

Answer 83

@beccaboo333 could you take a look at this please?