Equation of a circle passing through three points $(1,3),(-1,2),(2,1)$

Question

I cant get the final answer of this equation..i tried lot of things but i didnt get the right answer...how to solve this using the equation of a circle passing through three points?

Answer 1

Solving for the center $(a, b)$ and the radius $R$ with 3 equations may be a little tiring, so I suggest a more gradual approach:

1. Find the equations of 2 chords formed from the following points.

2. Find the equations of the perpendicular bisectors of the chords.

3. Solve for the center by finding the intersection of the perpendicular bisectors.

4. Solve for the radius by finding the distance between one of the given points and the center.

The equation of the circle is then

$$(x-a)^2+(y-b)^2=R^2$$

Answer 2

Hint:

Notice that a triangle is right at $(1,3)$ so midpoint of a segment between $A(-1,2),B(2,1)$ is a center of a circle and $2r =AB=\sqrt{10}$.