SOLUTION: determine the equation of the circle whose center is at (4,5) and tangent to the circle whose equation is x^2+y^2+4x+6y-23=0.

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Question 534153: determine the equation of the circle whose center is at (4,5) and tangent to the circle whose equation is x^2+y^2+4x+6y-23=0.
Found 2 solutions by Alan3354, KMST:
Answer by Alan3354(69443) About Me  (Show Source):
You can put this solution on YOUR website!
determine the equation of the circle whose center is at (4,5) and tangent to the circle whose equation is x^2+y^2+4x+6y-23=0
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Find the center of x^2+y^2+4x+6y-23=0
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x^2+y^2+4x+6y = 23
x%5E2+%2B+4x+%2B+4+%2B+y%5E2+%2B+6y+%2B+9+=+23+%2B+4+%2B+9+=+36
%28x%2B2%29%5E2+%2B+%28y%2B3%29%5E2+=+6%5E2
Center at (-2,-3), r = 6
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Find the distance from the center to (4,5)
d+=+sqrt%28diffy%5E2+%2B+diffx%5E2%29+=+sqrt%288%5E2+%2B+6%5E2%29+=+10
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The radius of the 2nd circle is 4, 10-6
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--> %28x-4%29%5E2+%2B+%28y-5%29%5E2+=+16

Answer by KMST(5328) About Me  (Show Source):
You can put this solution on YOUR website!
The given circle equation can be written as
+x%5E2%2By%5E2%2B4x%2B6y=23
and completing squares as
+x%5E2%2B4x%2B4%2By%5E2%2B6y%2B9=23%2B4%2B9 or %28x%2B2%29%5E2%2B%28y%2B3%29%5E2=36
showing that its radius is 6 and its center is (-2, -3).
The distance between the centers is

If the circles meet in between the centers (they are externally tangent), the radius of the second circle will be
10-6=4
and the equation for the second circle will be
%28x-4%29%5E2%2B%28y-5%29%5E2=16
which can be written as
+x%5E2%2By%5E2-8x-10y%2B25=0
If we make the second circle contain the first one (internally tangent circles), then the radius would be
10%2B6=16
and the equation for the second circle would be
%28x-4%29%5E2%2B%28y-5%29%5E2=256
which can be written as
+x%5E2%2By%5E2-8x-10y-215=0