Find the equation of the circle passing through the points $ (1,2) $, $ (3, -4) $, and $ (5, 6) $.

To find the equation of the circle passing through three points, we use the general form of the equation of a circle:

$ (x – h)^2 + (y – k)^2 = r^2 $

We substitute each point into the equation to get three equations with variables $ h $, $ k $, and $ r $:

$ (1 – h)^2 + (2 – k)^2 = r^2 \ (3 – h)^2 + (-4 – k)^2 = r^2 \ (5 – h)^2 + (6 – k)^2 = r^2 $

Solving these three equations simultaneously gives us the values of $ h $, $ k $, and $ r $. The final equation is:

$ (x – h)^2 + (y – k)^2 = r^2 $

To find the circle

Using the points $(1,2)$, $(3, -4)$, and $(5, 6)$, we solve the system:

$ (1 – h)^2 + (2 – k)^2 = r^2 (3 – h)^2 + (-4 – k)^2 = r^2 (5 – h)^2 + (6 – k)^2 = r^2 $

The circle

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