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# Two identical circles touch at the point P(9,3)

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Two identical circles touch at the point P(9,3)

one of the circles has equation x^2 + y^2 - 10x - 6y +18 = 0

find the equation of the other circle

Mar 22, 2021

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x^2  + y^2  - 10x  - 6y  + 18  = 0    complete the  square  on x, y

x^2  - 10y + 25  + y^2 - 6y + 9  =   -18  + 25 + 9      simplify

(x - 5)^2  +  ( y - 3)^2  =  16

The  center of this circle is   ( 5 , 3)    and the radius is  4

Then if  the  circles touch at   (9,3)  and they are identical....then  this  will be  the midpoint of  both centers

So.....let  the  center of the other circle  be (h, k)..so we have

[ 5 + h) / 2 =   9                  (3 + k) / 2  =  3

5 + h  =  9*2                       3 + k  =   3*2

5 + h = 18                           3 + k  = 6

h  =13                                     k = 3

The equation of  the second circle  is   ( x  -13)^2  + ( y - 3)^2  = 16

Here's a graph  :  https://www.desmos.com/calculator/efoxyb9720

Mar 22, 2021