How to find the equation of circle that passes through ($5,3$) , ($7,-2$) and ($-4,4$) circle with center at origin ($0,0$) and radius $r$?

Question

It is a challenge assignment on our class and I can't figure out how to solve it I always got stuck it is not the same as the other examples which are easy to solve. thanks

Answer 1

You know that the equation for a circle centered at $(0,0)$ with radius $r$ is $$x^2 + y^2 = r^2.$$Take any point, say $(5,3)$ and plug it into our equation and solve for $r$: $$(5)^2 + (3)^2 = 25 + 9 = 34 = r^2 \implies r = \sqrt{34}.$$ We need to verify this relationship with another point, say $(7,-2)$. So, $$(7)^2 + (-2)^2 = 49 + 4 = 53 = r^2 \implies r = \sqrt{53}$$ We have contradicting radii. So it is not possible for a circle to pass through all three of these points to be centered at the origin.

Answer 2

The three radii are different, so no origin centered circle can be defined.