What is the equation of the circle? How would i find it?

Question

Given three noncollinear​ points, there is one and only one circle that passes through them. Knowing that the equation of a circle may be written in the following​ form, find the equation of the circle passing through the given​ points, (0,4),(6,3)(-7,-3)

x^2+y^2+ax+by+c=0

the equation of the circle is x^2+y^2+()x+()y+()=0

Answer 1

(0,4), (6,3), and (-7,-3) are points on a circle

find the circle's equation

it helps to plot the 3 points and estimate a circle through them

then with algebra

plug in each point into

(x-h)^2 + (y-k)^2 = r^2

then solve the resulting system of equations for (h,k) and r

h^2 + (4-k)^2 = r^2

(6-h)^2 + (3-k)^2 = r^2

(-7-h)^2 + (-3-k)^2 = r^2

3 equations 3 unknowns

solve by substitution elimination

then convert to x^2 +y^2 + ax +by + c = 0 form

Answer 2

Substitute the three points into the equation getting three equatios in three variables.

Then solve the system by any method.