Question

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

Answer 1

Let point $(1,2)$ be $A$ , $(3,-4)$ be $B$ and $(5,-6)$ be $C$

Let the mid point of $AB$ be $D=(2,-1)$ and the midpoint of $BC$ be $E=(4,-5)$

The equation of line passing through $D$ and perpendicular to $AB$ is $x-3y=5$

The equation of line passing through $E$ and perpendicular to $BC$ is $x-y=9$

The center of circle is the point of intersection of above two lines which is equal to $(11,2)$

The radius of circle is $10$

Therefore the equation of circle is ${(x-11)}^2+{(y-2)}^2=100$