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August 7th, 2016, 06:02 AM   #1
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Standard Equation of Circle HELP!!

What is the standard form equation if concentric with x2 + y2 −8x−10y = −16 and 4 times the area ?
Explain how you got the answer too please..
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August 7th, 2016, 07:03 AM   #2
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You do understand that we give HELP, not answers, correct?

The general formula for a circle on a Cartesian plane is

$(x - h)^2 + (y - k)^2 = r^2$.

where (h, k) is the center and r is the radius. Obviously the equation you were given is NOT in the proper form.

The general method involved in solving questions like the one you were given is called completing the square.

$(u \pm v)^2 = u^2 \pm 2uv + v^2.$

Now suppose you have $u^2 \pm bu.$ That is NOT in the form of a square.

You could make it more like the form of the square if you re-expressed it this way:

$u^2 \pm bu = u^2 \pm 2 \left( \dfrac{b}{2} \right)u.$

But you need a second squared term to have it in the form of a square. So add zero in the form of the needed square term and its additive inverse.

$u^2 \pm 2 \left( \dfrac{b}{2} \right)u = u^2 \pm 2 \left( \dfrac{b}{2} \right)u + 0 = u^2 \pm 2 \left( \dfrac{b}{2} \right)u + \left \{ \left ( \dfrac{b}{2} \right )^2 - \left ( \dfrac{b}{2} \right )^2 \right \}.$

Well I have the extra term I wanted, but I still do not have the form of a square anywhere. So re-arrange the brackets to get the brackets in the right arrangement.

$u^2 \pm 2 \left( \dfrac{b}{2} \right)u + \left \{ \left ( \dfrac{b}{2} \right )^2 - \left ( \dfrac{b}{2} \right )^2 \right \} = \left \{u^2 \pm 2 \left( \dfrac{b}{2} \right)u + \left ( \dfrac{b}{2} \right )^2 \right \} - \left ( \dfrac{b}{2} \right )^2.$

Now I have a proper form for a square inside the brackets.

$\left \{u^2 \pm 2 \left( \dfrac{b}{2} \right)u + \left ( \dfrac{b}{2} \right )^2 \right \} - \left ( \dfrac{b}{2} \right )^2 = \left ( u \pm \dfrac{b}{2} \right )^2 - \dfrac{b^2}{4}.$ DONE.
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August 7th, 2016, 07:21 AM   #3
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ooh~~ Sorry~~ Im actually collecting some tricky questions on internet about our topic right now on Pre Calculus. And when I actually can't answer a problem that's when I ask for help here

Sorry and thanks too
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August 7th, 2016, 07:21 AM   #4
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Completing the Square: Circle Equations
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August 7th, 2016, 07:35 AM   #5
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Quote:
Originally Posted by skeeter View Post
Oh! Thanks for this!! really need something like this
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August 7th, 2016, 12:34 PM   #6
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x2 + y2 −8x−10y = −16

Rearrange & add 16 to both sides:
x2 −8x + 16 + y2 −10y = 0

Simplify into binomial square.
(x - 4)^2 + y2 - 10y = 0

Add 25 to both sides.
(x - 4)^2 + y2 - 10y + 25 = 25

Simplify into binomial square! We're Done!
(x - 4)^2 + (y - 5)^2 = 25

Not sure if this was the same thing that JeffM1 said but this is the simple algebraic approach.
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