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Find the radius of the circle with equation 9x^2-18x+9y^2+36y+44=0.

 May 15, 2019
 #1
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Let's get the equation into the form

 

(x - h)2 + (y - k)2  =  r2     where the point  (h, k)  is the center of the circle and  r  is the radius.

 

9x2 - 18x + 9y2 + 36y + 44  =  0

                                                         Subtract  44  from both sides of the equation

9x2 - 18x + 9y2 + 36y  =  -44

                                                         Divide both sides by  9

x2 - 2x + y2 + 4y  =  - \(\frac{44}{9}\)

                                                         Add  1  and add  4  to both sides to complete the squares on the left side

 

x2 - 2x + 1  +  y2 + 4y + 4   =   - \(\frac{44}{9}\) + 1 + 4

                                                                         Factor both perfect square trinomials on the left side

(x - 1)2  +  (y + 2)2   =   - \(\frac{44}{9}\) + 1 + 4

                                                                         Get a common denominator to combine  - \(\frac{44}{9}\) + 1 + 4

(x - 1)2  +  (y + 2)2   =   - \(\frac{44}{9}\) + \(\frac99\) + \(\frac{36}{9}\)

 

(x - 1)2  +  (y + 2)2   =   \(\frac19\)

 

Now it is in the form     (x - h)2 + (y - k)2  =  r2     and we can see that...

 

r2  =  \(\frac19\)

                  The radius is positive so take the positive sqrt of both sides

r  =  \(\frac13\)

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 May 15, 2019

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