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The parabola y = x^2 is tangent to a circle at (2,4).  The circle passes through the point (0,7).  Find the center of the circle.

 Feb 16, 2022
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The center of the circle will be on the  y axis.....let's call this point (0 , a)

 

The distance from this point to  (2,4)   and to (0,7)  will be the radius

 

So  we  can equate distances   (using the square of the distance formula on  both sides of the  equation )

 

( 0 - 0 )^2  +  ( 7 -a)^2  =  ( 2- 0)^2  + ( 4 - a)^2          simplify

 

     a2  - 14a  + 49   =  4  + a^2  - 8a  + 16 

 

         -14a +  49  =  -8a + 20

 

           49 -20  =  14a - 8a

 

               29  =   6a

 

                a = 29/6 

 

 

 

So the center is ( 0, 29/6)  and the radius =  7 - 29/6   =   42/6 - 29/6 =   13/6  

 

The radius  =   7 - 29/6  =   42/6 - 29/6 =  13/6

 

 

cool cool cool

 Feb 16, 2022

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