What is the standard form of the equation for this circle?

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What is the standard form of the equation for this circle?

Mathematics
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@hotguy @Vocaloid @ any one who can help me
The standard form of a circle is \((x-h)^2+(y-k)^2=r^2\) where the center of the circle is \((h, k)\) and \(r\) is the radius. Can you take it from there?

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okay so then that would mean it is (x-4)^2 + (y+5)^2 = 5.5^2
right?
Yup. Great job!
thanks!
np :)
@Calcmathlete could you help me with another question?
Sure
The equation of a circle is x2 + y2 + Cx + Dy + E = 0. If the radius of the circle is decreased without changing the coordinates of the center point, how are the coefficients C, D, and E affected? i dont get how to work with circles
Do you know how to complete the square?
no
Ok, completing the square is kind of a technique that is similar to factoring (somewhat). It's based on the formula \[(a\pm b)^2=a^2\pm2ab+b^2\] The idea is that you're trying to get a quadratic into a form that can become a perfect square. Before we work on your question, let's work with the idea first.\[x^2-4x+5=10\] Now, of course, the easiest thing would be to factor, but let's try completing the square instead.\[x^2-4x+\_\_=5\] You're trying to find the number that should go there to make it a perfect square trinomial.\[x^2-4x\color{red}{+4}=5\color{red}{+4}\]We added 4 on the left side to make it the perfect square trinomial (this is to make it fit the form above). However, to balance the equation, we need to add 4 to the right side as well.\[(x-2)^2=9\]That is called completing the square. Does that make sense so far?
kind of, yes
okay now what?
Alright, so what we need to do is complete the square for the equation of the circle above. \[x^2+y^2+Cx+Dy+E=0\]\[(x^2+Cx+\_\_)+(y^2+Dy+\_\_)=-E\]\[\left(x^2+Cx+\left(\frac{C}{2}\right)^2\right)+\left(y^2+Dy+\left(\frac D2\right)^2\right)=-E+\left(\frac C2\right)^2+\left(\frac D2\right)^2\]\[\left(x+\frac C2\right)^2+\left(y+\frac D2\right)^2=-E+\left(\frac C2\right)^2+\left(\frac D2\right)^2\]This should look familiar from the form above.\[(x-h)^2+(y-k)^2=r^2\]where \(h=-\frac C2\), \(k=-\frac D2\), \(r^2=-E+\left(\frac C2\right)^2+\left(\frac D2\right)^2\)
If you're not quite 100% on completing the square yet, ^ is going to be a bit hard to follow.
okay i kinda follow its a bit hazy but its there
Alright, I'll go over completing the square in a sec again, but to finish up your question, the variables that affect the center of the circle are C and D. Your question asked that if r decreased, how would C, D, E be affected if the center of the circle didn't change. So, since the center didn't move, C and D stay constant. However, E needs to increase for the radius to decrease. Does that make sense? I'll go back to completing the square after this.
okay yeah that does
the only problem is thats not an answer
What is the answer according to your source?
these are our options
With the way the question is written, I don't think any of the choices are correct. Are you sure that's the exact wording of the question?
yeah i copy and paste it
I just double checked everything on my paper. I think their question is just straight up wrong. If it were - E or something instead of + E, then the choice would be on there. I'd just go with the last option since it's the closest one, but...wow...
i chose e
http://www.wolframalpha.com/input/?i=x%5E2%2By%5E2%2B6x%2B6y%2B5%3D0 http://www.wolframalpha.com/input/?i=x%5E2%2By%5E2%2B6x%2B6y%2B10%3D0 Also, ^ that is why I'm sure that their question is flawed.
The +10 one is clearly smaller than the +5 one, and the centers stayed the same.
Sorry for all the confusion.
its okay thanks tho
np

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